Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
Energy Technology Data Exchange (ETDEWEB)
Dou, Nicholas G.; Minnich, Austin J. [Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125 (United States)
2016-01-04
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
A generalized linear Boltzmann equation for non-classical particle transport
International Nuclear Information System (INIS)
This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications.
Punshon-Smith, Samuel; Smith, Scott
2016-01-01
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kin...
Transition flow ion transport via integral Boltzmann equation
International Nuclear Information System (INIS)
A new approach is developed to solve the Integral Boltzmann Equation for the evolving velocity distribution of a source of ions, undergoing electrostatic acceleration through a neutral gas target. The theory is applicable to arbitrarily strong electric fields, any ion/neutral mass ratio greater than unity, and is not limited to spatially isotropic gas targets. A hard sphere collision model is used, with a provision for inelasticity. Both axial and radial velocity distributions are calculated for applications where precollision radial velocities are negligible, as is the case for ion beam extractions from high pressure sources. Theoretical predictions are tested through an experiment in which an atmospheric pressure ion source is coupled to a high vacuum energy analyser. Excellent agreement results for configurations in which the radial velocity remains small. Velocity distributions are applied to predicting the efficiency of coupling an atmospheric pressure ion source to a quadrupole mass spectrometer and results clearly indicate the most desirable extracting configuration. A method is devised to calculate ion-molecule hard sphere collision cross sections for easily fragmented organic ions
International Nuclear Information System (INIS)
Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the PN approximation, the SN method (discrete ordinates method) and their variants. In the discrete ordinates (SN) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to SN two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the PN approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented. (author)
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Energy Technology Data Exchange (ETDEWEB)
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
International Nuclear Information System (INIS)
The Aron equation is a generalization of the Fokker-Planck equation allowing for diffusion motion with finite maximal velocity. The Aron equation can be regarded as a semi-phenomenological equation because it is based on phenomenological laws such as the Fick diffusion law. It is shown that the one-dimensional case of the Aron equation can be derived from the Boltzmann transport equation for particles in Zitterbewegung. The extension to the three-dimensional case, however, leads to an equation different from the Aron one
A simple Boltzmann transport equation for ballistic to diffusive transient heat transport
International Nuclear Information System (INIS)
Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions
International Nuclear Information System (INIS)
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano’s theorem. Additionally, Lewis’ approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano’s and Lewis’ approaches are stated in this new equation. Fano’s theorem is found not to apply in the presence of electromagnetic fields. Lewis’ theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms. (paper)
Tervo, J; Frank, M; Herty, M
2016-01-01
The paper considers a coupled system of linear Boltzmann transport equation (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems for numerical methods. First, we prove the existence and uniqueness of solutions, under sufficient criteria and in appropriate $L^2$-based spaces, of a single (particle) CSDA-equation by using two complementary techniques, the Lions-Lax-Milgram Theorem (variational approach), and the theory evolution operators (semigroup approach). The necessary a priori estimates are shown. In addition, we prove the corresponding results and estimates for the system of coupled transport equat...
Cobos, Agustín C.; Poma, Ana L.; Alvarez, Guillermo D.; Sanz, Darío E.
2016-10-01
We introduce an alternative method to calculate the steady state solution of the angular photon flux after a numerical evolution of the time-dependent Boltzmann transport equation (BTE). After a proper discretization the transport equation was converted into an ordinary system of differential equations that can be iterated as a weighted Richardson algorithm. As a different approach, in this work the time variable regulates the iteration process and convergence criteria is based on physical parameters. Positivity and convergence was assessed from first principles and a modified Courant-Friedrichs-Lewy condition was devised to guarantee convergence. The Penelope Monte Carlo method was used to test the convergence and accuracy of our approach for different phase space discretizations. Benchmarking was performed by calculation of total fluence and photon spectra in different one-dimensional geometries irradiated with 60Co and 6 MV photon beams and radiological applications were devised.
Takane, Yositake; Hayashi, Masahiko; Ebisawa, Hiromichi
2016-08-01
The time-dependent Ginzburg-Landau equation and the Boltzmann transport equation for charge-density-wave (CDW) conductors are derived from a microscopic one-dimensional model by applying the Keldysh Green's function approach under a quasiclassical approximation. The effects of an external electric field and impurity pinning of the CDW are fully taken into account without relying on a phenomenological argument. These equations simultaneously describe the spatiotemporal dynamics of both the CDW and quasiparticles; thus, they serve as a starting point to develop a general framework to analyze various nonequilibrium phenomena, such as current conversion between the CDW condensate and quasiparticles, in realistic CDW conductors. It is shown that, in typical situations, the equations correctly describe the nonlinear behavior of electric conductivity in a simpler manner.
Radtke, Gregg A; Hadjiconstantinou, Nicolas G
2009-05-01
We present an efficient variance-reduced particle simulation technique for solving the linearized Boltzmann transport equation in the relaxation-time approximation used for phonon, electron, and radiative transport, as well as for kinetic gas flows. The variance reduction is achieved by simulating only the deviation from equilibrium. We show that in the limit of small deviation from equilibrium of interest here, the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium, in stark contrast to standard particle simulation methods. Our results demonstrate that a space-dependent equilibrium distribution improves the variance reduction achieved, especially in the collision-dominated regime where local equilibrium conditions prevail. We also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation-time approximation, a very simple collision algorithm with a clear physical interpretation can be formulated. PMID:19518597
An introduction to the theory of the Boltzmann equation
Harris, Stewart
2011-01-01
Boltzmann's equation (or Boltzmann-like equations) appears extensively in such disparate fields as laser scattering, solid-state physics, nuclear transport, and beyond the conventional boundaries of physics and engineering, in the fields of cellular proliferation and automobile traffic flow. This introductory graduate-level course for students of physics and engineering offers detailed presentations of the basic modern theory of Boltzmann's equation, including representative applications using both Boltzmann's equation and the model Boltzmann equations developed within the text. It emphasizes
Energy Technology Data Exchange (ETDEWEB)
Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx
2003-07-01
There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)
Romano, Giuseppe; Esfarjani, Keivan; Strubbe, David A.; Broido, David; Kolpak, Alexie M.
2016-01-01
Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of phonons, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann transport equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute 17 % to heat transport, compared to 5 % in bulk Si. Interestingly, we observe a constant thermal conductivity over the range 200 K acoustic phonons with long intrinsic MFP and the temperature dependence of the heat capacity. Our findings, which are in qualitative agreement with the temperature trend of thermal conductivities measured in nanoporous Si-based systems, shed light on the origin of the reduction of thermal conductivity in nanostructured materials and demonstrate the necessity of multiscale heat transport engineering, in which the bulk material and geometry are optimized concurrently.
Inelastic Quantum Transport in Superlattices: Success and Failure of the Boltzmann Equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, Stephan;
1999-01-01
the whole held range from linear response to negative differential conductivity. The quantum results are compared with the respective results obtained from a Monte Carlo solution of the Boltzmann equation. Our analysis thus sets the limits of validity for the semiclassical theory in a nonlinear...
International Nuclear Information System (INIS)
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
Fiorentini, Mattia; Bonini, Nicola
2016-08-01
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to n -doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.
Kinetic Boltzmann, Vlasov and Related Equations
Sinitsyn, Alexander; Vedenyapin, Victor
2011-01-01
Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in
Energy Technology Data Exchange (ETDEWEB)
Lloyd, S. A. M.; Ansbacher, W. [Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada); Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada) and Department of Medical Physics, British Columbia Cancer Agency-Vancouver Island Centre, Victoria, British Columbia V8R 6V5 (Canada)
2013-01-15
Purpose: Acuros external beam (Acuros XB) is a novel dose calculation algorithm implemented through the ECLIPSE treatment planning system. The algorithm finds a deterministic solution to the linear Boltzmann transport equation, the same equation commonly solved stochastically by Monte Carlo methods. This work is an evaluation of Acuros XB, by comparison with Monte Carlo, for dose calculation applications involving high-density materials. Existing non-Monte Carlo clinical dose calculation algorithms, such as the analytic anisotropic algorithm (AAA), do not accurately model dose perturbations due to increased electron scatter within high-density volumes. Methods: Acuros XB, AAA, and EGSnrc based Monte Carlo are used to calculate dose distributions from 18 MV and 6 MV photon beams delivered to a cubic water phantom containing a rectangular high density (4.0-8.0 g/cm{sup 3}) volume at its center. The algorithms are also used to recalculate a clinical prostate treatment plan involving a unilateral hip prosthesis, originally evaluated using AAA. These results are compared graphically and numerically using gamma-index analysis. Radio-chromic film measurements are presented to augment Monte Carlo and Acuros XB dose perturbation data. Results: Using a 2% and 1 mm gamma-analysis, between 91.3% and 96.8% of Acuros XB dose voxels containing greater than 50% the normalized dose were in agreement with Monte Carlo data for virtual phantoms involving 18 MV and 6 MV photons, stainless steel and titanium alloy implants and for on-axis and oblique field delivery. A similar gamma-analysis of AAA against Monte Carlo data showed between 80.8% and 87.3% agreement. Comparing Acuros XB and AAA evaluations of a clinical prostate patient plan involving a unilateral hip prosthesis, Acuros XB showed good overall agreement with Monte Carlo while AAA underestimated dose on the upstream medial surface of the prosthesis due to electron scatter from the high-density material. Film measurements
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
Coarse-grained transport of a turbulent flow via moments of the Reynolds-averaged Boltzmann equation
Abramov, Rafail V
2015-01-01
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by the Reynolds average of the fine-grained global relative entropy, and thus obeys the time decay bound of Desvillettes and Villani. This is similar to what is observed in the rarefied gas dynamics, which makes the Grad moment closure a good candidate for truncating the hierarchy of the coarse-grained moment equations. We also show that, under additional assumptions on the form of the coarse-grained collision terms, one a...
Cekmen, Z. C.; Dincer, M. S.
2009-07-01
The effective ionization coefficients and transport parameters such as electron mean energy drift velocity and transverse diffusion coefficient in binary and ultradilute SF6-Ar gas mixtures have been calculated for density reduced electric field strength E/N values from 10 to 400 Td. These calculations have been performed by using the two-term spherical harmonic expansion to obtain the numerical solution of the Boltzmann transport equation based on the finite element method under steady-state Townsend condition. In order to confirm the model and code developed in this study, the Reid ramp model has been used as a benchmark test and then effective ionization coefficients and transport parameters have been evaluated for SF6 contents of 1%, 10%, 25%, 50%, 70% and 100% in the binary mixture. Finally SF6 contents in the ultradilute mixtures of 0.1%, 0.3%, 0.5% and 0.7% are taken into account with the evaluated effective ionizations and transport parameters of electron mean energy, drift velocity and transverse diffusion coefficients.
Goffin, Mark A.; Baker, Christopher M. J.; Buchan, Andrew G.; Pain, Christopher C.; Eaton, Matthew D.; Smith, Paul N.
2013-06-01
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k with directional dependence. General error estimators are derived for any given functional of the flux and applied to k to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.
Energy Technology Data Exchange (ETDEWEB)
Bankovic, A., E-mail: ana.bankovic@gmail.com [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Dujko, S. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Centrum Wiskunde and Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam (Netherlands); ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); White, R.D. [ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); Buckman, S.J. [ARC Centre for Antimatter-Matter Studies, Australian National University, Canberra, ACT 0200 (Australia); Petrovic, Z.Lj. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia)
2012-05-15
This work reports on a new series of calculations of positron transport properties in molecular hydrogen under the influence of spatially homogeneous electric field. Calculations are performed using a Monte Carlo simulation technique and multi term theory for solving the Boltzmann equation. Values and general trends of the mean energy, drift velocity and diffusion coefficients as a function of the reduced electric field E/n{sub 0} are reported here. Emphasis is placed on the explicit and implicit effects of positronium (Ps) formation on the drift velocity and diffusion coefficients. Two important phenomena arise; first, for certain regions of E/n{sub 0} the bulk and flux components of the drift velocity and longitudinal diffusion coefficient are markedly different, both qualitatively and quantitatively. Second, and contrary to previous experience in electron swarm physics, there is negative differential conductivity (NDC) effect in the bulk drift velocity component with no indication of any NDC for the flux component. In order to understand this atypical manifestation of the drift and diffusion of positrons in H{sub 2} under the influence of electric field, the spatially dependent positron transport properties such as number of positrons, average energy and velocity and spatially resolved rate for Ps formation are calculated using a Monte Carlo simulation technique. The spatial variation of the positron average energy and extreme skewing of the spatial profile of positron swarm are shown to play a central role in understanding the phenomena.
Energy Technology Data Exchange (ETDEWEB)
St Aubin, J., E-mail: joel.st.aubin@albertahealthservices.ca; Keyvanloo, A.; Fallone, B. G. [Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue Northwest, Edmonton, Alberta T6G 1Z2 (Canada); Vassiliev, O. [Department of Medical Physics, Tom Baker Cancer Center, 1331 29 Street Northwest, Calgary, Alberta T2N 4N2 (Canada)
2015-02-15
Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm{sup 2} field as well as a smaller 2 × 2 cm{sup 2} field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization
International Nuclear Information System (INIS)
The fundamental process for determining the electric discharge phenomena of gas which take various forms depending on the kinds of gas, gas pressure, relative position of electrodes and applied voltage, is the mutual collision of electrons, atoms, molecular ions and neutral atoms and molecules. The initial stage before the establishment of electric discharge seems to be in Townsend discharge region where the collision of electrons with neutral molecules and atoms mainly occurs, being the weakly ionized condition at low gas temperature. Recently, the breakdown phenomena of N2-O2 gas mixture is being examined for the purpose of clarifying the impulse break mechanism in low pressure air, and the energy distribution of electrons and the transport coefficients in N2, O2 and N2-O2 mixed gases are required to investigate closely the results. Here, the energy distribution and the transport coefficients of electrons in steady Townsend discharge region in N2 and O2 gases respectively were analyzed by using Boltzmann equation, as a preparatory stage. The analyzed results and the discussions on them are reported in each paragraph of the energy distribution and the mean energy of electrons, ionization coefficients and adhesion coefficients, electron drift speed and diffusion coefficients, and excitation frequencies for various electron levels. It was confirmed that each collision process for electrons and the cross-section used for the analysis were properly selected. The excitation frequencies of electrons for N2 and O2 gases concerning the band spectra emitted from discharge channels and the electron energy distribution at 200 V/cm-Torr or below of E/P0 were newly calculated, where E is electric field, and P0 is gas pressure at 0 deg C. (Wakatsuki, Y.)
Priimak, Dmitri
2014-01-01
We present finite differences numerical algorithm for solving 2D spatially homogeneous Boltzmann transport equation for semiconductor superlattices (SL) subject to time dependant electric field along SL axis and constant perpendicular magnetic field. Algorithm is implemented in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPUs. We compare performance and merits of one implementation versus another and discuss various methods of optimization.
International Nuclear Information System (INIS)
Purpose: To evaluate the dose distributions of an 192Ir source (model VS2000) in homogeneous water geometry calculated using a deterministic grid-based Boltzmann transport equation solver (GBBS) in the commercial treatment planning system (TPS) (BRACHYVISION-ACUROS v8.8). Methods: Using percent dose differences (%ΔD), the GBBS (BV-ACUROS) was compared to the (1) published TG-43 data, (2) MCNPX Monte Carlo (MC) simulations of the 192Ir source centered in a 15 cm radius water sphere, and (3) TG-43 output from the TPS using vendor supplied (BV-TG43-vendor) and user extended (BV-TG43-extended) 2D anisotropy functions F(r,θ). BV-ACUROS assumes 1 mm of NiTi cable, while the TPS TG-43 algorithm uses data based on a 15 cm cable. MC models of various cable lengths were simulated. Results: The MC simulations resulted in >20% dose deviations along the cable for 1, 2, and 3 mm cable lengths relative to 15 cm. BV-ACUROS comparisons with BV-TG43-vendor and BV-TG43-extended yielded magnitude of differences, consistent with those seen in MC simulations. However, differences >20% extended further (θ≤10 deg.) when using the vendor supplied anisotropy function Fven(r,θ). These differences were also seen in comparisons of F(r,θ) derived from the TPS output. Conclusions: The results suggest that %ΔD near the cable region is larger than previously estimated. The spatial distribution of the dose deviation is highly dependent on the reference TG-43 data used to compare to GBBS. The differences observed, while important to realize, should not have an impact on clinical dosimetry in homogeneous water.
de Urquijo, Jaime; Basurto, E.; Juarez, A. M.; Ness, Kevin; Robson, Robert; Brunger, Michael; White, Ron
2014-10-01
The drift velocity of electrons in mixtures of gaseous water with helium and argon are measured over the range of reduced electric fields from 0--300 Td using a pulsed-Townsend technique. Small admixtures of water to both helium and argon are found to produce negative differential conductivity (NDC), despite NDC being absent from the pure gases. Comparison of the measured drift velocities with those calculated from a multi-term solution of Boltzmann's equation provides a further discriminative assessment on the accuracy and completeness of electron water vapour cross-sections. Funding acknowledgements: ARC, Mexican govt (PAPIIT IN 111014).
The Boltzmann equation in the difference formulation
Energy Technology Data Exchange (ETDEWEB)
Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
THREE WAY DECOMPOSITION FOR THE BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
Ilgis Ibragimov; Sergej Rjasanow
2009-01-01
The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.
Lattice Boltzmann Model and Geophysical Hydrodynamic Equation
Institute of Scientific and Technical Information of China (English)
冯士德; 杨京龙; 郜宪林; 季仲贞
2002-01-01
A lattice Boltzmann equation model in a rotating system is developed by introducing the Coriolis force effect.The geophysical hydrodynamic equation can be derived from this model. Numerical computations are performed to simulate the cylindrical annulus experiment and Benard convection. The numerical results have shown the flow behaviour of large-scale geostrophic current and Benard convection cells, which verifies the applicability of this model to both theory and experiment.
Test of Information Theory on the Boltzmann Equation
Hyeon-Deuk, Kim; Hayakawa, Hisao
2002-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
Test of Information Theory on the Boltzmann Equation
Kim, Hyeon-Deuk; Hayakawa, Hisao
2003-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-09-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
Wagner, Alexander J
2016-01-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
Energy Technology Data Exchange (ETDEWEB)
Stoenescu, M.L.
1977-06-01
The terms in Boltzmann kinetic equation corresponding to elastic short range collisions, inelastic excitational collisions, coulomb interactions and electric field acceleration are evaluated numerically for a standard distribution function minimizing the computational volume by expressing the terms as linear combinations with recalculable coefficients, of the distribution function and its derivatives. The present forms are suitable for spatial distribution calculations.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Asymptotic-preserving Boltzmann model equations for binary gas mixture
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations.
Analysis of Jeans instability from Boltzmann equation
Kremer, Gilberto M
2015-01-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. The equilibrium distribution function takes into account the expansion of the Universe and a pressureless fluid in the matter dominated Universe. Without invoking Jeans "swindle" a dispersion relation is obtained by considering small perturbations of the equilibrium values of the distribution function and gravitational potential. The collapse criterion -- which happens in an unstable region where the solution grows exponentially with time -- is determined from the dispersion relation. The collapse criterion in a static Universe occurs when the wavenumber $k$ is smaller than the Jeans wavenumber $k_J$, which was the solution found by Jeans. For an expanding Universe it is shown that this criterion is $k\\leq\\sqrt{7/6}\\,k_J$. As a consequence the ratio of the mass contained in a sphere of diameter equal to the wavelength $\\lambda=2\\pi/k$ to t...
Supersymmetric electroweak baryogenesis, nonequilibrium field theory and quantum Boltzmann equations
Riotto, Antonio
1998-01-01
The closed time-path (CPT) formalism is a powerful Green's function formulation to describe nonequilibrium phenomena in field theory and it leads to a complete nonequilibrium quantum kinetic theory. In this paper we make use of the CPT formalism to write down a set of quantum Boltzmann equations describing the local number density asymmetries of the particles involved in supersymmetric electroweak baryogenesis. These diffusion equations automatically and self-consistently incorporate the CP-violating sources which fuel baryogenesis when transport properties allow the CP-violating charges to diffuse in front of the bubble wall separating the broken from the unbroken phase at the electroweak phase transition. This is a significant improvement with respect to recent approaches where the CP-violating sources are inserted by hand into the diffusion equations. Furthermore, the CP-violating sources and the particle number changing interactions manifest ``memory'' effects which are typical of the quantum transp ort t...
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
Thermal equation of state for lattice Boltzmann gases
Institute of Scientific and Technical Information of China (English)
Ran Zheng
2009-01-01
The Galilean invaxiance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model axe proposed together with their rigorous theoretical background. From the viewpoint of group invariance,recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
Thermal equation of state for lattice Boltzmann gases
Ran, Zheng
2009-06-01
The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
A probabilistic view on the general relativistic Boltzmann equation
Bailleul, Ismael
2011-01-01
A new probalistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal character of the relativistic Boltzmann model.
Langevin theory of fluctuations in the discrete Boltzmann equation
Gross, M; Varnik, F; Adhikari, R
2010-01-01
The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuation-dissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, the fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equations of fluctuating hydrodynamics for ideal and non-ideal fluids. Application of the framework to a generic force-based non-ideal fluid model leads to ideal gas-type thermal noise. Simulation results indicate proper thermalization of all degrees of freedom.
Philippi, P C; Surmas, R; Philippi, Paulo Cesar; Santos, Luis Orlando Emerich dos; Surmas, Rodrigo
2005-01-01
The particles model, the collision model, the polynomial development used for the equilibrium distribution, the time discretization and the velocity discretization are factors that let the lattice Boltzmann framework (LBM) far away from its conceptual support: the continuous Boltzmann equation (BE). Most collision models are based on the BGK, single parameter, relaxation-term leading to constant Prandtl numbers. The polynomial expansion used for the equilibrium distribution introduces an upper-bound in the local macroscopic speed. Most widely used time discretization procedures give an explicit numerical scheme with second-order time step errors. In thermal problems, quadrature did not succeed in giving discrete velocity sets able to generate multi-speed regular lattices. All these problems, greatly, difficult the numerical simulation of LBM based algorithms. In present work, the systematic derivation of lattice-Boltzmann models from the continuous Boltzmann equation is discussed. The collision term in the li...
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Metamaterial characterization using Boltzmann's kinetic equation for electrons
DEFF Research Database (Denmark)
Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.;
2013-01-01
Statistical properties of electrons in metals are taken into consideration to describe the microscopic motion of electrons. Assuming degenerate electron gas in metal, we introduce the Boltzmann kinetic equation to supplement Maxwell's equations. The solution of these equations clearly shows the...
Celebrating Cercignani's conjecture for the Boltzmann equation
Villani, Cédric
2011-01-01
Cercignani\\'s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann\\'s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s. © American Institute of Mathematical Sciences.
Spinor Boltzmann Equation with Two Momenta at the Fermi Level
Institute of Scientific and Technical Information of China (English)
王正川
2012-01-01
Based on the formalism of Keldysh's nonequilibrium Green function, we establish a two momenta spinor Boltzmann equation for longitudinal scalar distribution function and transverse vector distribution function. The lon- gitudinal charge currents, transverse spin currents and the continuity equations satisfied by them are then studied, it indicates that both the charge currents and spin currents decay oscillately along with position, which is due to the momenta integral over the Fermi surface. We also compare our charge currents and spin currents with the corresponding results of one momentum spinor Boltzmann equation, the differences are obvious.
Celebrating Cercignani's conjecture for the Boltzmann equation
Desvillettes, Laurent; Villani, Cédric
2010-01-01
Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.
Thermal creep problems by the discrete Boltzmann equation
Directory of Open Access Journals (Sweden)
L. Preziosi
1991-05-01
Full Text Available This paper deals with an initial-boundary value problem for the discrete Boltzmann equation confined between two moving walls at different temperature. A model suitable for the quantitative analysis of the initial boundary value problem and the relative existence theorem are given.
An exactly solvable non-linear Boltzmann equation
Ernst, M.H.; Hendriks, E.M.
1979-01-01
The initial value problem for a model Boltzmann equation of a two dimensional gas with a continuous or discrete energy distribution function and a transition probability δ(ε - ε') is solved exactly; ε and ε' are the total energies before and after collision.
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Modeling adsorption with lattice Boltzmann equation
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
CORRECTIONS TO THE COLLISION TERM IN THE BGK BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; REN RONG-CAI; CUI XIAO-PENG; JI ZHONG-ZHEN
2001-01-01
With the discrete method of the hexagonal cell and three different velocities of particle population in each cell,a two-dimensional lattice Boltzmann model is developed in this paper.[1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.[3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q19 model is verified.
The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
Vasques, Richard
2015-01-01
We show that, by correctly selecting the probability distribution function $p(s)$ for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of $p(s)$ preserves the $true$ mean-squared free path of the system, which sheds new light on the results obtained in previous work.
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E.; Niemi, H.; Rischke, D. H.
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break dow...
Shock-wave structure using nonlinear model Boltzmann equations.
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
Transport Equations for Oscillating Neutrinos
Zhang, Yunfan
2013-01-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse su...
Non-linear effects in the Boltzmann equation
International Nuclear Information System (INIS)
The Boltzmann equation is studied by defining an integral transformation of the energy distribution function for an isotropic and homogeneous gas. This transformation may be interpreted as a linear superposition of equilibrium states with variable temperatures. It is shown that the temporal evolution features of the distribution function are determined by the singularities of said transformation. This method is applied to Maxwell and Very Hard Particle interaction models. For the latter, the solution of the Boltzmann equation with the solution of its linearized version is compared, finding out many basic discrepancies and non-linear effects. This gives a hint to propose a new rational approximation method with a clear physical meaning. Applying this technique, the relaxation features of the BKW (Bobylev, Krook anf Wu) mode is analyzed, finding a conclusive counter-example for the Krook and Wu conjecture. The anisotropic Boltzmann equation for Maxwell models is solved as an expansion in terms of the eigenfunctions of the corresponding linearized collision operator, finding interesting transient overpopulation and underpopulation effects at thermal energies as well as a new preferential spreading effect. By analyzing the initial collision, a criterion is established to deduce the general features of the final approach to equilibrium. Finally, it is shown how to improve the convergence of the eigenfunction expansion for high energy underpopulated distribution functions. As an application of this theory, the linear cascade model for sputtering is analyzed, thus finding out that many differences experimentally observed are due to non-linear effects. (M.E.L.)
LATTICE BOLTZMANN EQUATION MODEL IN THE CORIOLIS FIELD
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; MAO JIANG-YU; ZHANG QIONG
2001-01-01
In a large-scale field of rotational fluid, various unintelligible and surprising dynamic phenomena are produced due to the effect of the Coriolis force. The lattice Boltzmann equation (LBE) model in the Coriolis field is developed based on previous works.[1-4] Geophysical fluid dynamics equations are derived from the model. Numerical simulations have been made on an ideal atmospheric circulation of the Northern Hemisphere by using the model and they reproduce the Rossby wave motion well. Hence the applicability of the model is verified in both theory and experiment.
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnár, Etele; Niemi, Harri; Rischke, Dirk H.
2016-06-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, f^0 k, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from f^0 k. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics.
On the Krook-Wu model of the Boltzmann equation
Cornille, H.
1980-08-01
The distribution function of the Krook-Wu model of the nonlinear Boltzmann equation (elastic differential cross sections inversely proportional to the relative speed of the colliding particles) is obtained as a generalized Laguerre polynomial expansion where the only time dependence is provided by the coefficients. In a recent paper M. Barnsley and the present author have shown that these coefficients are recursively determined from the resolution of a nonlinear differential system. Here we explicitly show how to construct the solutions of the Krook-Wu model and study the properties of the corresponding Krook-Wu distribution functions.
A new lattice Boltzmann equation to simulate density-driven convection of carbon dioxide
Allen, Rebecca
2013-01-01
The storage of CO2 in fluid-filled geological formations has been carried out for more than a decade in locations around the world. After CO2 has been injected into the aquifer and has moved laterally under the aquifer\\'s cap-rock, density-driven convection becomes an important transport process to model. However, the challenge lies in simulating this transport process accurately with high spatial resolution and low CPU cost. This issue can be addressed by using the lattice Boltzmann equation (LBE) to formulate a model for a similar scenario when a solute diffuses into a fluid and density differences lead to convective mixing. The LBE is a promising alternative to the traditional methods of computational fluid dynamics. Rather than discretizing the system of partial differential equations of classical continuum mechanics directly, the LBE is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. We propose an extension to the LBE, which can accurately predict the transport of dissolved CO2 in water, as a step towards fluid-filled porous media simulations. This is achieved by coupling two LBEs, one for the fluid flow and one for the convection and diffusion of CO2. Unlike existing lattice Boltzmann equations for porous media flow, our model is derived from a system of moment equations and a Crank-Nicolson discretization of the velocity-truncated Boltzmann equation. The forcing terms are updated locally without the need for additional central difference approximation. Therefore our model preserves all the computational advantages of the single-phase lattice Boltzmann equation and is formally second-order accurate in both space and time. Our new model also features a novel implementation of boundary conditions, which is simple to implement and does not suffer from the grid-dependent error that is present in the standard "bounce-back" condition. The significance of using the LBE in this work lies in the ability to efficiently
Phonon Boltzmann equation-based discrete unified gas kinetic scheme for multiscale heat transfer
Guo, Zhaoli
2016-01-01
Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant-Friedrichs-Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.
Asinari, P.
2011-03-01
Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others. PMID:27627421
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br, E-mail: ftvdl@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica. Grupo de Pesquisas Radiologicas; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: phd.marcio@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte
2015-07-01
This work presents a new analytical method for solving the Boltzmann equation. In this formulation, a linear differential operator is applied over the Boltzmann model, in order to produce a partial differential equation in which the scattering term is absent. This auxiliary equation is solved via reduction of order. The exact solution obtained is employed to define a precursor for the buildup factor. (author)
Boltzmann Equation Solver Adapted to Emergent Chemical Non-equilibrium
Birrell, Jeremiah
2014-01-01
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space occupation factor $\\Upsilon(t)$. In this first paper we address (effectively) massless fermions and derive dynamical equations for $T(t)$ and $\\Upsilon(t)$ such that the zeroth order term of the basis alone captures the number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component ($e^\\pm$-annihilation).
Nonaligned shocks for discrete velocity models of the Boltzmann equation
Directory of Open Access Journals (Sweden)
J. M. Greenberg
1991-05-01
Full Text Available At the conclusion of I. Bonzani's presentation on the existence of structured shock solutions to the six-velocity, planar, discrete Boltzmann equation (with binary and triple collisions, Greenberg asked whether such solutions were possible in directions e(α=(cosα ,sinα when α was not one of the particle flow directions. This question generated a spirited discussion but the question was still open at the conclusion of the conference. In this note the author will provide a partial resolution to the question raised above. Using formal perturbation arguments he will produce approximate solutions to the equation considered by Bonzani which represent traveling waves propagating in any direction e(α=(cosα ,sinα.
Neutron transport equation - indications on homogenization and neutron diffusion
International Nuclear Information System (INIS)
In PWR nuclear reactor, the practical study of the neutrons in the core uses diffusion equation to describe the problem. On the other hand, the most correct method to describe these neutrons is to use the Boltzmann equation, or neutron transport equation. In this paper, we give some theoretical indications to obtain a diffusion equation from the general transport equation, with some simplifying hypothesis. The work is organised as follows: (a) the most general formulations of the transport equation are presented: integro-differential equation and integral equation; (b) the theoretical approximation of this Boltzmann equation by a diffusion equation is introduced, by the way of asymptotic developments; (c) practical homogenization methods of transport equation is then presented. In particular, the relationships with some general and useful methods in neutronic are shown, and some homogenization methods in energy and space are indicated. A lot of other points of view or complements are detailed in the text or the remarks
Stamnes, K.; Lie-Svendsen, O.; Rees, M. H.
1991-01-01
The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E; Rischke, D H
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. To zeroth order this expansion yields ideal fluid dynamics, to first order Navier-Stokes theory, and to second order transient theories of dissipative fluid dynamics. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, so-called anisotropic fluid dynamics, in terms of an expansion around a single-particle distribution function which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. In this paper we derive, up to terms of second order in this expansion, the equations of mo...
High order numerical methods for the space non-homogeneous Boltzmann equation
International Nuclear Information System (INIS)
In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods
A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
ZHANGChao-Ying; TANHui-Li; LIUMu-Ren; KONGLing-Jiang
2004-01-01
A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.
Lattice Boltzmann Hydrodynamic and Transport Modeling of Everglades Mangrove Estuaries
Sukop, M. C.; Engel, V.
2010-12-01
Lattice Boltzmann methods are being developed and applied to simulate groundwater and surface water flows, and heat, solute, and particle transport. Their ability to solve Navier-Stokes, St. Venant, or Darcy equations with closely coupled solute transport and density-dependent flow effects in geometrically complex domains is attractive for inverse modeling of tracer release data and forward modeling of carbon transport in mangrove estuaries under various future conditions. Key physical processes to be simulated include tidal cycles, storm surge, sea level change, variable upstream stage, subsurface groundwater inputs, and precipitation/recharge and their effects on estuary salinity and carbon transport in the estuaries and groundwater beneath the mangroves. Carbon sources and storage in the aquifer and exchanges at the mangrove-estuary interface and carbon transformations in the water column also need to be simulated. Everglades tidal mangrove estuaries are characterized by relatively high velocity (approaching 1 m s-1) tidal flows. The channels are generally less than 2 m in depth. Tidal fluctuations approach 2 m leading to significant areas of periodic inundation and emergence of oyster beds, shell beaches, mangrove root masses, and sandy beaches. Initial models are two-dimensional, although a three-dimensional model explicitly incorporating bathymetry, density-dependent flow, and wind-driven circulation could be developed. Preliminary work highlights some of the abilities of early models. A satellite image of a 64-km2 area surrounding a CO2 flux tower is used to provide the model geometry. Model resolution is 15 m per grid node. A sinusoidal tidal stage variation and constant, high salinity are applied to the Gulf side of the model while a constant stage (corresponding to mean tide), zero salinity boundary is applied on the inland side. The Navier-Stokes equations coupled with the advection-diffusion equation are solved in the open channels. The mangrove areas
On the solution of ion transport equation
International Nuclear Information System (INIS)
Two different techniques have been used to solve the Boltzmann-transport equation describing the penetration of light ions through solids in the continuous slowing down approximation (CSDA), namely maximum entropy and flux-limited.The solution- obtained for the scalar flux function θο(ξ,s) by using the flux-limited as well by means of the maximum entropy are agree with obtained by Chandrasekhar method
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project
National Aeronautics and Space Administration — The overall objective of the proposed project is to develop a generalized lattice Boltzmann (GLB) approach as a potential computational aeroacoustics (CAA) tool for...
Equivalence Between Forward and Backward Boltzmann Equations in Multi-Component Medium
Institute of Scientific and Technical Information of China (English)
张竹林
2002-01-01
The author generalized the propagator function theory introduced first by Sigmund, and gave a explicitly proof of a equivalence between forward and backward Boltzmann equations in a multi-component medium by using the generalized propagator function theory.
Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity
Shi-you Lin
2014-01-01
The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the ${C}^{\\infty }$ solutions in non-Maxwellian and strong singularity cases.
Note on Invariance of One-Dimensional Lattice-Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
RAN Zheng
2007-01-01
Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background.It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation (LBE),especially for shock calculation.Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.
THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE
Institute of Scientific and Technical Information of China (English)
Yuanjie LEI
2014-01-01
This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1-3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].
Simulation of a Natural Convection by the Hybrid Thermal Lattice Boltzmann Equation
Energy Technology Data Exchange (ETDEWEB)
Ryu, Seungyeob; Kang, Hanok; Seo, Jaekwang; Yun, Juhyeon; Zee, Sung-Quun [Korea Atomic Energy Research Institute, Taejon (Korea, Republic of)
2006-07-01
Recently, the lattice Boltzmann method(LBM) has gained much attention for its ability to simulate fluid flows, and for its potential advantages over conventional CFD method. The key advantages of LBM are, (1) suitability for parallel computations, (2) absence of the need to solve the time-consuming Poisson equation for a pressure, and (3) an ease with multiphase flows, complex geometries and interfacial dynamics may be treated. In spite of its success in solving various challenging problems involving athermal fluids, the LBM has not been able to handle realistic thermal fluids with a satisfaction. The difficulty encountered in the thermal LBM seems to be the numerical instabilities. The existing thermal lattice Boltzmann models may be classified into three categories based on their approach in solving the Boltzmann equation, namely, the multispeed, the passive scalar and the thermal energy distribution approach. For more details see Ref. In the present work, the hybrid thermal lattice Boltzmann scheme proposed by Lallemand and Luo is used for simulating a natural convection in a square cavity. They proposed a hybrid thermal lattice Boltzmann equation(HTLBE) in which the mass and momentum conservation equations are solved by using the multiple-relaxation-time(MRT) model, whereas the diffusion-advection equations for the temperature are solved separately by using finite-difference technique. The main objective of the present work is to establish the lattice Boltzmann method as a viable tool for the simulation of temperature fields at high Rayleigh numbers.
Electronic transport in VO{sub 2}—Experimentally calibrated Boltzmann transport modeling
Energy Technology Data Exchange (ETDEWEB)
Kinaci, Alper; Rosenmann, Daniel; Chan, Maria K. Y., E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439 (United States); Kado, Motohisa [Higashifuji Technical Center, Toyota Motor Corporation, Susono, Shizuoka 410-1193 (Japan); Ling, Chen; Zhu, Gaohua; Banerjee, Debasish, E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Materials Research Department, Toyota Motor Engineering and Manufacturing North America, Inc., Ann Arbor, Michigan 48105 (United States)
2015-12-28
Materials that undergo metal-insulator transitions (MITs) are under intense study, because the transition is scientifically fascinating and technologically promising for various applications. Among these materials, VO{sub 2} has served as a prototype due to its favorable transition temperature. While the physical underpinnings of the transition have been heavily investigated experimentally and computationally, quantitative modeling of electronic transport in the two phases has yet to be undertaken. In this work, we establish a density-functional-theory (DFT)-based approach with Hubbard U correction (DFT + U) to model electronic transport properties in VO{sub 2} in the semiconducting and metallic regimes, focusing on band transport using the Boltzmann transport equations. We synthesized high quality VO{sub 2} films and measured the transport quantities across the transition, in order to calibrate the free parameters in the model. We find that the experimental calibration of the Hubbard correction term can efficiently and adequately model the metallic and semiconducting phases, allowing for further computational design of MIT materials for desirable transport properties.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region
Lattice Boltzmann Model for The Volume-Averaged Navier-Stokes Equations
Zhang, Jingfeng; Ouyang, Jie
2014-01-01
A numerical method, based on discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.
Molnár, Etele; Rischke, Dirk H
2016-01-01
In Moln\\'ar et al. [Phys. Rev. D 93, 114025 (2016)] the equations of anisotropic dissipative fluid dynamics were obtained from the moments of the Boltzmann equation based on an expansion around an arbitrary anisotropic single-particle distribution function. In this paper we make a particular choice for this distribution function and consider the boost-invariant expansion of a fluid in one dimension. In order to close the conservation equations, we need to choose an additional moment of the Boltzmann equation. We discuss the influence of the choice of this moment on the time evolution of fluid-dynamical variables and identify the moment that provides the best match of anisotropic fluid dynamics to the solution of the Boltzmann equation in the relaxation-time approximation.
Normal and adjoint integral and integrodifferential neutron transport equations. Pt. 2
International Nuclear Information System (INIS)
Using the simplifying hypotheses of the integrodifferential Boltzmann equations of neutron transport, given in JEN 334 report, several integral equations, and theirs adjoint ones, are obtained. Relations between the different normal and adjoint eigenfunctions are established and, in particular, proceeding from the integrodifferential Boltzmann equation it's found out the relation between the solutions of the adjoint equation of its integral one, and the solutions of the integral equation of its adjoint one (author)
Weak and strong coupling limits of the Boltzmann equation in the relaxation-time approximation
Jaiswal, Amaresh; Redlich, Krzysztof
2016-01-01
We consider a momentum dependent relaxation time for the Boltzmann equation in the relaxation time approximation. We employ a power law parametrization for the momentum dependence of the relaxation time, and calculate the shear and bulk viscosity, as well as, the charge and heat conductivity. We show, that for the two popular parametrizations, referred to as the linear and quadratic ansatz, one can obtain transport coefficients which corresponds to the weak and strong coupling regimes, respectively. We also show that, for a system of massless particles with vanishing chemical potential, the off-equilibrium corrections to the phase-space distribution function calculated with the quadratic ansatz are identical with those of the Grad's 14-moment method.
Bazow, D; Heinz, U; Martinez, M; Noronha, J
2016-01-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of non-hydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the non-hydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation...
Directory of Open Access Journals (Sweden)
Nilson C. Roberty
2011-01-01
Full Text Available We introduce algorithms marching over a polygonal mesh with elements consistent with the propagation directions of the particle (radiation flux. The decision for adopting this kind of mesh to solve the one-speed Boltzmann transport equation is due to characteristics of the domain of the transport operator which controls derivatives only in the direction of propagation of the particles (radiation flux in the absorbing and scattering media. This a priori adaptivity has the advantages that it formulates a consistent scheme which makes appropriate the application of the Lax equivalence theorem framework to the problem. In this work, we present the main functional spaces involved in the formalism and a description of the algorithms for the mesh generation and the transport equation solution. Some numerical examples related to the solution of a transmission problem in a high-contrast model with absorption and scattering are presented. Also, a comparison with benchmarks problems for source and reactor criticality simulations shows the compatibility between calculations with the algorithms proposed here and theoretical results.
Discrete Boltzmann model of shallow water equations with polynomial equilibria
Meng, Jianping; Emerson, David R; Peng, Yong; Zhang, Jianmin
2016-01-01
A hierarchy of discrete Boltzmann model is proposed for simulating shallow water flows. By using the Hermite expansion and Gauss-Hermite quadrature, the conservation laws are automatically satisfied without extra effort. Moreover, the expansion order and quadrature can be chosen flexibly according to the problem for striking the balance of accuracy and efficiency. The models are then tested using the classical one-dimensional dam-breaking problem, and successes are found for both supercritical and subcritical flows.
A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off
Silvestre, Luis
2016-11-01
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in {L^∞} which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Energy Technology Data Exchange (ETDEWEB)
Lee, B
2008-12-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving S{sub n} discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
Entropy inequality and hydrodynamic limits for the Boltzmann equation.
Saint-Raymond, Laure
2013-12-28
Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!). PMID:24249776
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
Liu, Shuangqian; Yang, Xiongfeng
2016-08-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
Mahakrishnan, Sathiya; Chakraborty, Subrata; Vijay, Amrendra
2016-09-15
Diffusion, an emergent nonequilibrium transport phenomenon, is a nontrivial manifestation of the correlation between the microscopic dynamics of individual molecules and their statistical behavior observed in experiments. We present a thorough investigation of this viewpoint using the mathematical tools of quantum scattering, within the framework of Boltzmann transport theory. In particular, we ask: (a) How and when does a normal diffusive transport become anomalous? (b) What physical attribute of the system is conceptually useful to faithfully rationalize large variations in the coefficient of normal diffusion, observed particularly within the dynamical environment of biological cells? To characterize the diffusive transport, we introduce, analogous to continuous phase transitions, the curvature of the mean square displacement as an order parameter and use the notion of quantum scattering length, which measures the effective interactions between the diffusing molecules and the surrounding, to define a tuning variable, η. We show that the curvature signature conveniently differentiates the normal diffusion regime from the superdiffusion and subdiffusion regimes and the critical point, η = ηc, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example
Mahakrishnan, Sathiya; Chakraborty, Subrata; Vijay, Amrendra
2016-09-15
Diffusion, an emergent nonequilibrium transport phenomenon, is a nontrivial manifestation of the correlation between the microscopic dynamics of individual molecules and their statistical behavior observed in experiments. We present a thorough investigation of this viewpoint using the mathematical tools of quantum scattering, within the framework of Boltzmann transport theory. In particular, we ask: (a) How and when does a normal diffusive transport become anomalous? (b) What physical attribute of the system is conceptually useful to faithfully rationalize large variations in the coefficient of normal diffusion, observed particularly within the dynamical environment of biological cells? To characterize the diffusive transport, we introduce, analogous to continuous phase transitions, the curvature of the mean square displacement as an order parameter and use the notion of quantum scattering length, which measures the effective interactions between the diffusing molecules and the surrounding, to define a tuning variable, η. We show that the curvature signature conveniently differentiates the normal diffusion regime from the superdiffusion and subdiffusion regimes and the critical point, η = ηc, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example
Peristaltic particle transport using the Lattice Boltzmann method
Energy Technology Data Exchange (ETDEWEB)
Connington, Kevin William [Los Alamos National Laboratory; Kang, Qinjun [Los Alamos National Laboratory; Viswanathan, Hari S [Los Alamos National Laboratory; Abdel-fattah, Amr [Los Alamos National Laboratory; Chen, Shiyi [JOHNS HOPKINS UNIV.
2009-01-01
Peristaltic transport refers to a class of internal fluid flows where the periodic deformation of flexible containing walls elicits a non-negligible fluid motion. It is a mechanism used to transport fluid and immersed solid particles in a tube or channel when it is ineffective or impossible to impose a favorable pressure gradient or desirous to avoid contact between the transported mixture and mechanical moving parts. Peristaltic transport occurs in many physiological situations and has myriad industrial applications. We focus our study on the peristaltic transport of a macroscopic particle in a two-dimensional channel using the lattice Boltzmann method. We systematically investigate the effect of variation of the relevant dimensionless parameters of the system on the particle transport. We find, among other results, a case where an increase in Reynolds number can actually lead to a slight increase in particle transport, and a case where, as the wall deformation increases, the motion of the particle becomes non-negative only. We examine the particle behavior when the system exhibits the peculiar phenomenon of fluid trapping. Under these circumstances, the particle may itself become trapped where it is subsequently transported at the wave speed, which is the maximum possible transport in the absence of a favorable pressure gradient. Finally, we analyze how the particle presence affects stress, pressure, and dissipation in the fluid in hopes of determining preferred working conditions for peristaltic transport of shear-sensitive particles. We find that the levels of shear stress are most hazardous near the throat of the channel. We advise that shear-sensitive particles should be transported under conditions where trapping occurs as the particle is typically situated in a region of innocuous shear stress levels.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng
2014-01-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 x 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng; Mendl, Christian B.
2015-06-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 × 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as a special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation
Arkeryd, Leif; Nouri, Anne
2015-01-01
A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two-dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value. 1 Anyons and the Boltzmann equation. Let us first recall the definition of anyon. Con...
A Fokker-Planck model of the Boltzmann equation with correct Prandtl number
Mathiaud, J
2015-01-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
Xia, Yin; Xu, Jun; Li, Bao-An; Shen, Wen-Qing
2016-08-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
Chen, Li; Zhang, Lei; Tao, Wenquan
2014-01-01
Size, morphology and distributions of pores in organic matters of shale matrix are discussed based on high resolution images from experiments in the literature. 150 nanoscale structures of the organic matters are then reconstructed by randomly placing pore spheres with different diameters and overlap tolerances. Effects of porosity, the mean diameter and the overlap tolerance on void space connectivity and pore size distribution are studied. Further, a pore-scale model based on the Lattice Boltzmann method is developed to predict the Knudsen diffusivity and permeability of the reconstructed organic matters. The simulation results show that the mean pore diameter and overlap tolerance significantly affect the transport properties. The predicted Knudsen effective diffusivity is compared with Bruggeman equation and it is found that this equation underestimate the tortuosity. A modified Bruggeman equation is proposed based on the simulation results. The predicted intrinsic permeability is in acceptable agreement ...
Generalized Boltzmann equations for on-shell particle production in a hot plasma
Jakovác, A
2002-01-01
A novel refinement of the conventional treatment of Kadanoff--Baym equations is suggested. Besides the Boltzmann equation another differential equation is used for calculating the evolution of the non-equilibrium two-point function. Although it was usually interpreted as a constraint on the solution of the Boltzmann equation, we argue that its dynamics is relevant to the determination and resummation of the particle production cut contributions. The differential equation for this new contribution is illustrated in the example of the cubic scalar model. The analogue of the relaxation time approximation is suggested. It results in the shift of the threshold location and in smearing out of the non-analytic threshold behaviour of the spectral function. Possible consequences for the dilepton production are discussed.
Monitoring derivation of the quantum linear Boltzmann equation
Hornberger, Klaus; Vacchini, Bassano
2007-01-01
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced in [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a non-perturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear...
General particle transport equation. Final report
International Nuclear Information System (INIS)
The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence
Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force
Institute of Scientific and Technical Information of China (English)
Seiji UKAI; Tong YANG; Huijiang ZHAO
2006-01-01
For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.
Stability of Global Solution to Boltzmann-Enskog Equation with External Force
Institute of Scientific and Technical Information of China (English)
JIANG ZHENG-LU; MA LI-JUN; YAO ZHENG-AN
2012-01-01
In the presence of external forces depending only on the time and space variables,the Boltzmann-Enskog equation formally conserves only the mass of the system,and its entropy functional is also nonincreasing.Corresponding to this type of equation,we first give some hypotheses of its bicharacteristic equations and then get some results about the stablity of its global solution with the help of two new Lyapunov functionals:one is to describe interactions between particles with different velocities and the other is to measure the L1 distance between two mild solutions.The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the L1 stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.
From Newton's law to the linear Boltzmann equation without cut-off
Ayi, Nathalie
2016-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cutoff starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combin...
A hybrid kinetic-fluid model for solving the gas dynamics Boltzmann-BGK equation
Crouseilles, Nicolas; Degond, Pierre; Lemou, Mohammed
2004-01-01
International audience Our purpose s toderive a hybrid model for particles systems which combines a kinetic description of the fast particles with a fluid description of the thermal ones. Fats particles will be described through a collisional kinetic equation of Boltzmann-BGK type while thermal particles will be modeled by means of a system of a Euler type equations. A conservative numerical scheme is constructed and enables us to validate the approach on various numerical tests.
Steady detonation waves via the Boltzmann equation for a reacting mixture
Conforto, F; Schürrer, F; Ziegler, I
2003-01-01
Based on the Boltzmann equation, the detonation problem is dealt with on a mesoscopic level. The model is based on the assumption that ahead of a shock an explosive gas mixture is in meta stable equilibrium. Starting from the Von Neumann point the chemical reaction, initiated by the pressure jump, proceeds until the chemical equilibrium is reached. Numerical solutions of the derived macroscopic equations as well as the corresponding Hugoniot diagrams which reveal the physical relevance of the mathematical model are provided.
Formulating Weak Lensing from the Boltzmann Equation and Application to Lens-lens Couplings
Su, S -C
2014-01-01
The Planck mission has conclusively detected lensing of the Cosmic Microwave Background (CMB) radiation from foreground sources to an overall significance of greater than $25\\sigma$. The high precision of this measurement motivates the development of a more complete formulation of the calculation of this effect. While most effects on the CMB anisotropies are widely studied through direct solutions of the Boltzmann equation, the non-linear effect of CMB lensing is formulated through the solutions of the geodesic equation. In this paper, we present a new formalism to the calculation of the lensing effect by \\emph{directly solving the Boltzmann equation}, as we did in the calculation of the CMB anisotropies at recombination. In particular, we developed a diagrammatic approach to efficiently keep track of all the interaction terms and calculate all possible non-trivial correlations to arbitrary high orders. Using this formalism, we explicitly articulate the approximations required to recover the usual remapping a...
Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
Abramov, Rafail V
2016-01-01
In the current work we propose a diffusive modification of the Boltzmann equation. This naturally leads to the corresponding diffusive fluid dynamics equations, which we numerically investigate in a simple Couette flow setting. This diffusive modification is based on the assumption of the "imperfect" model collision term, which is unable to track all collisions in the corresponding real gas particle system. The effect of missed collisions is then modeled by an appropriately scaled long-term homogenization process of the particle dynamics. The corresponding diffusive fluid dynamics equations are produced in a standard way by closing the hierarchy of the moment equations using either the Euler or the Grad closure. In the numerical experiments with the Couette flow, we discover that the diffusive Euler equations behave similarly to the conventional Navier-Stokes equations, while the diffusive Grad equations additionally exhibit Knudsen-like velocity boundary layers. We compare the simulations with the correspond...
A novel protocol for linearization of the Poisson-Boltzmann equation
Tsekov, R
2014-01-01
A new protocol for linearization of the Poisson-Boltzmann equation is proposed and the resultant electrostatic equation coincides formally with the Debye-Huckel equation, the solution of which is well known for many electrostatic problems. The protocol is examined on the example of electrostatically stabilized nano-bubbles and it is shown that stable nano-bubbles could be present in aqueous solutions of anionic surfactants near the critical temperature, if the surface potential is constant. At constant surface charge non nano-bubbles could exist.
Global solutions in the critical Besov space for the non-cutoff Boltzmann equation
Morimoto, Yoshinori; Sakamoto, Shota
2016-10-01
The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyze the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained.
Asinari, Pietro
2010-01-01
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both ...
Premnath, Kannan N; Banerjee, Sanjoy
2008-01-01
Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment-projections of source/forcing terms are derived such that they recover preconditioned Navier-Stokes equations with non-uniform external forces. As an illustration, we solve an extende...
Xie, Dexuan; Jiang, Yi
2016-10-01
The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20]. As the development of this recent work, in this paper, a nonlocal modified Poisson-Boltzmann equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson-Boltzmann equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.
Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
Holst, Michael; Yu, Zeyun; Zhou, Yongcheng; Zhu, Yunrong
2010-01-01
We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a priori pointwise estimates and the first complete solution and approximation theory for the Poisson-Boltzmann equation. It also made possible the first provably convergent discretization of the PBE, and allowed for the development of a provably convergent AFEM for the PBE. However, in practice the regularization turns out to be numerically ill-conditioned. In this article, we examine a second regularization, and establish a number of basic results to ensure that the new approach produces the same mathematical advantages of the original regularization, without the ill-conditioning property. We then design an AFEM scheme based on the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This res...
Energy Technology Data Exchange (ETDEWEB)
EL Safadi, M
2007-03-15
We study the regularity of kinetic equations of Boltzmann type.We use essentially Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadics annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff). In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C{sup {infinity}} regularity. Then, we deal with the case of general cross-sections with 'hard potential'. We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones. Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation. (author)
Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson-Walker space-time
Takou, Etienne
2016-01-01
In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson-Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space.
Sliding periodic boundary conditions for lattice Boltzmann and lattice kinetic equations
Adhikari, R.; Desplat, J. -C.; Stratford, K.
2005-01-01
We present a method to impose linear shear flow in discrete-velocity kinetic models of hydrodynamics through the use of sliding periodic boundary conditions. Our method is derived by an explicit coarse-graining of the Lees-Edwards boundary conditions for Couette flow in molecular dynamics, followed by a projection of the resulting equations onto the subspace spanned by the discrete velocities of the lattice Boltzmann method. The boundary conditions are obtained without resort to perturbative ...
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
Lahanas, A B; Nanopoulos, Dimitri V
2006-01-01
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
Lahanas, Ab; Mavromatos, Ne; Nanopoulos, Dv
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
José Colmenares; Antonella Galizia; Jesús Ortiz; Andrea Clematis; Walter Rocchia
2014-01-01
The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is ...
Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation
Institute of Scientific and Technical Information of China (English)
LI; Zhihui; ZHANG; Hanxin; FU; Song
2005-01-01
The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.
Biro, T. S.; Molnar, E.
2011-01-01
We derive equations for fluid dynamics from a non-extensive Boltzmann transport equation consistent with Tsallis' non-extensive entropy formula. We evaluate transport coefficients employing the relaxation time approximation and investigate non-extensive effects in leading order dissipative phenomena at relativistic energies, like heat conductivity, shear and bulk viscosity.
Coclite, Alessandro; Pascazio, Giuseppe; Decuzzi, Paolo
2016-01-01
Modelling the vascular transport and adhesion of man-made particles is crucial for optimizing their efficacy in the detection and treatment of diseases. Here, a Lattice Boltzmann and Immersed Boundary methods are combined together for predicting the near wall dynamics of particles with different shapes in a laminar flow. For the lattice Boltzmann modelling, a Gauss-Hermite projection is used to derive the lattice equation, wall boundary conditions are imposed through the Zou-He framework, and a moving least squares algorithm accurately reconstructs the forcing term accounting for the immersed boundary. First, the computational code is validated against two well-known test cases: the sedimentation of circular and elliptical cylinders in a quiescent fluid. A very good agreement is observed between the present results and those available in the literature. Then, the transport of circular, elliptical, rectangular, square and triangular particles is analyzed in a Couette flow, at Re=20. All particles drifted later...
Dyatko, Nikolay; Donko, Zoltan
2015-01-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This "bistability effect" - in which electron-electron (Coulomb) collisions play an essential role - is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms ("two-term app...
Solution Poisson-Boltzmann equation: Application in the Human Neuron Membrane
Soares, M A G; Cortez, C M
2008-01-01
With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of Poisson-Boltzmann. In this work we consider these solutions for the membrane of the human neuron, using a model simplified for this structure considering the distribution of electrolytes in each side of the membrane, as well as the effect of glycocalyx and the lipidic bilayer. It was assumed that on both sides of the membrane the charges are homogeneously distributed and that the potential depends only on coordinate z.
A multi-term solution of the space-time Boltzmann equation for electrons in gaseous and liquid Argon
Boyle, G J; Tattersall, W J; McEachran, R P; White, R D
2015-01-01
In a recent paper [1] the scattering and transport of excess electrons in liquid argon in the hydrodynamic regime was investigated, generalizing the seminal works of Lekner and Cohen [2,3] with modern scattering theory techniques and kinetic theory. In this paper, the discussion is extended to the non-hydrodynamic regime through the development of a full multi-term space-time solution of Boltzmann's equation for electron transport in gases and liquids using a novel operator-splitting method. A Green's function formalism is considered that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the hydrodynamic regime is studied for a benchmark model Percus-Yevick liquid as well as for liquid argon. The temporal evolution of Franck-Hertz oscillations are observed for liquids, with striking differences in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations in arg...
Chatterjee, Dipankar; Amiroudine, Sakir
2011-02-01
A comprehensive non-isothermal Lattice Boltzmann (LB) algorithm is proposed in this article to simulate the thermofluidic transport phenomena encountered in a direct-current (DC) magnetohydrodynamic (MHD) micropump. Inside the pump, an electrically conducting fluid is transported through the microchannel by the action of an electromagnetic Lorentz force evolved out as a consequence of the interaction between applied electric and magnetic fields. The fluid flow and thermal characteristics of the MHD micropump depend on several factors such as the channel geometry, electromagnetic field strength and electrical property of the conducting fluid. An involved analysis is carried out following the LB technique to understand the significant influences of the aforementioned controlling parameters on the overall transport phenomena. In the LB framework, the hydrodynamics is simulated by a distribution function, which obeys a single scalar kinetic equation associated with an externally imposed electromagnetic force field. The thermal history is monitored by a separate temperature distribution function through another scalar kinetic equation incorporating the Joule heating effect. Agreement with analytical, experimental and other available numerical results is found to be quantitative.
Patel, R.A.; Perko, J.; Jaques, D.; De Schutter, G.; Ye, G.; Van Breugel, K.
2013-01-01
A Lattice Boltzmann (LB) based reactive transport model intended to capture reactions and solid phase changes occurring at the pore scale is presented. The proposed approach uses LB method to compute multi component mass transport. The LB multi-component transport model is then coupled with the well
Perturbative and non-perturbative aspects non-Abelian Boltzmann-Langevin equation
Energy Technology Data Exchange (ETDEWEB)
Boedeker, Dietrich. E-mail: bodeker@physik.uni-bielefeld.de
2002-12-30
We study the Boltzmann-Langevin equation which describes the dynamics of hot Yang-Mills fields with typical momenta of order of the magnetic screening scale g{sup 2}T. It is transformed into a path integral and Feynman rules are obtained. We find that the leading log Langevin equation can be systematically improved in a well behaved expansion in log(1/g){sup -1}. The result by Arnold and Yaffe that the leading log Langevin equation is still valid at next-to-leading-log order is confirmed. We also confirm their result for the next-to-leading-log damping coefficient, or color conductivity, which is shown to be gauge fixing independent for a certain class of gauges. The frequency scale g{sup 2}T does not contribute to this result, but it does contribute, by power counting, to the transverse gauge field propagator. Going beyond a perturbative expansion we find 1-loop ultraviolet divergences which cannot be removed by renormalizing the parameters in the Boltzmann-Langevin equation.
Fokker-Planck Equation for Boltzmann-type and Active Particles transfer probability approach
Trigger, S A
2002-01-01
Fokker-Planck equation with the velocity-dependent coefficients is considered for various isotropic systems on the basis of probability transition (PT) approach. This method provides the self-consistent and universal description of friction and diffusion for Brownian particles. Renormalization of the friction coefficient is shown to occur for two dimensional (2-D) and three dimensional (3-D) cases, due to the tensorial character of diffusion. The specific forms of PT are calculated for the Boltzmann-type of collisions and for the absorption-type of collisions (the later are typical for dusty plasmas and some other systems). Validity of the Einstein's relation for the Boltzmann-type collisions is proved for the velocity-dependent friction and diffusion coefficients. For non-Boltzmann collisions, such as, e.g., absorption collisions, the Einstein relation is violated, although some other relations (determined by the structure of PT) can exist. The collecting part of the ion drag force in a dusty plasma, arising...
Energy Technology Data Exchange (ETDEWEB)
Ayissi, Raoul Domingo, E-mail: raoulayissi@yahoo.fr; Noutchegueme, Norbert, E-mail: nnoutch@yahoo.fr [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
International Nuclear Information System (INIS)
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Ayissi, Raoul Domingo; Noutchegueme, Norbert
2015-01-01
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
Lu, Xuguang; Mouhot, Clément
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
Numerical solution of the Boltzmann equation for the shock wave in a gas mixture
Raines, A A
2014-01-01
We study the structure of a shock wave for a two-, three- and four-component gas mixture on the basis of numerical solution of the Boltzmann equation for the model of hard sphere molecules. For the evaluation of collision integrals we use the Conservative Projection Method developed by F.G. Tscheremissine which we extended to gas mixtures in cylindrical coordinates. The transition from the upstream to downstream uniform state is presented by macroscopic values and distribution functions. The obtained results were compared with numerical and experimental results of other authors.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.
Khurana, Saheba; Thachuk, Mark
2016-03-14
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation. PMID:26979675
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
Zheng, Lin; Zhai, Qinglan
2014-01-01
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface fore (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter visa Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is also solved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and a two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then ...
International Nuclear Information System (INIS)
In this paper we combine a stochastic 3D microstructure model of a fiber based gas diffusion layer of polymer electrolyte fuel cells with a Lattice Boltzmann model for fluid transport. We focus on a simple approach of compressing the planar oriented virtual geometry of paper-type gas diffusion layer from Toray. Material parameters – permeability and tortuosity – are calculated from simulation of one phase, one component gas flow in stochastic geometries. We analyze the statistical spread of simulation results on ensembles of the virtual geometry, both uncompressed and compressed. The influence of the compression is discussed with regard to the Kozeny–Carman equation. The effective transport properties calculated from transport simulations in compressed gas diffusion layers agree well with a trend based on the Kozeny–Carman equation
Düring, Bertram
2015-01-01
We propose and investigate different kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider:(i) opinion dynamics under the effect of opinion leadership, where each individual is characterised not only by its opinion, but also by another independent variable which quantifies leadership qualities; (ii) opinion dynamics modelling political segregation in the `The Big Sort', a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker-Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Directory of Open Access Journals (Sweden)
José Colmenares
2014-01-01
Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs. PMID:25013789
Onsager's-principle-consistent 13-moment transport equations.
Singh, Narendra; Agrawal, Amit
2016-06-01
A new set of generalized transport equations is derived for higher-order moments which are generated in evolution equation for stress tensor and heat flux vector in 13-moment equations. The closure we employ satisfies Onsager's symmetry principle. In the derivation, we do not employ a phase density function based on Hermite polynomial series in terms of higher-order moments, unlike Grad's approach. The distribution function is rather chosen to satisfy collision invariance, and H-theorem and capture relatively strong deviations from equilibrium. The phase density function satisfies the linearized Boltzmann equation and provides the correct value of the Prandtl number for monatomic gas. The derived equations are compared with Grad's 13-moments equations for gas modeled as Maxwellian molecule. The merits of the proposed equations against Grad's and R13 equations are discussed. In particular, it is noted that the proposed equations contain higher-order terms compared to these equations but require a fewer number of boundary conditions as compared to the R13 equations. The Knudsen number envelope which can be covered to describe flows with these equations is therefore expected to be larger as compared to the earlier equations.
LATTICE BOLTZMANN METHOD SIMULATIONS FOR MULTIPHASE FLUIDS WITH REDICH-KWONG EQUATION OF STATE
Institute of Scientific and Technical Information of China (English)
WEI Yi-kun; QIAN Yue-hong
2011-01-01
In this article we state that the compression factor of the Redlich-Kwong Equation Of State (EOS) is smaller than that of van der Waals EOS.The Redlich-Kwong EOS is in better agreement with experimental data on coexistence curves at the critical point than the van der Waals EOS.We implement the Redlich-Kwong EOS in the Lattice Boltzmann Method (LBM) simulations via a pseudo-potential approach.We propose a new force,which can obtain computational stationary and reach larger density ratio.As a result,multi-phase flows with large density ratio (up to 1012 in the stationary case) can be simulated.We perform four numerical simulations,which are respectively related to single liquid droplet,vapor-liquid separation,surface tension and liquid coalescence of two droplets.
Longaretti, Pierre-Yves
2016-01-01
These 1992 lectures notes present a powerful formalism mostly developed in the 1980s by Borderies, Goldreich and Tremaine to address planetary ring dynamical issues. These notes make a special emphasis on ring microphysics, quantified with the help of the moments of the Boltzmann equation. They also focus on the standard self-gravity model of narrow ring rigid precession, and on the physics of linear and nonlinear density waves. These notes have been corrected but only very marginally extended and not updated. They are provided both as an introduction to the streamline formalism and as a complement on some technical issues for my upcoming review ("Theory of Narrow rings and Sharp Edges") that will cover the physics not addressed here along with more recent developments. This review will appear in the "Planetary Ring System" book (C. Murray and M. Tiscareno, eds.), to be published later on this year at Cambridge University Press.
Hong, Liu; Yang, Zaibao; Zhu, Yi; Yong, Wen-An
2015-12-01
In this article, we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, properly separating entropy fluxes and production rates, and determining a dissipation matrix. Our approach takes advantage of both extended irreversible thermodynamics and GENERIC formalisms and shows a direct correspondence with Levermore's moment-closure hierarchies for the Boltzmann equation. As a direct application, a new ten-moment model beyond the classical hierarchies is constructed and is shown to recover the Euler equations in the equilibrium state. These interesting results may put various macroscopic modeling approaches, starting from the general principles of non-equilibrium thermodynamics, on a solid microscopic foundation based on the Boltzmann equation.
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution). PMID:24182245
Volume transport and generalized hydrodynamic equations for monatomic fluids.
Eu, Byung Chan
2008-10-01
In this paper, the effects of volume transport on the generalized hydrodynamic equations for a pure simple fluid are examined from the standpoint of statistical mechanics and, in particular, kinetic theory of fluids. First, we derive the generalized hydrodynamic equations, namely, the constitutive equations for the stress tensor and heat flux for a single-component monatomic fluid, from the generalized Boltzmann equation in the presence of volume transport. Then their linear steady-state solutions are derived and examined with regard to the effects of volume transport on them. The generalized hydrodynamic equations and linear constitutive relations obtained for nonconserved variables make it possible to assess Brenner's proposition [Physica A 349, 11 (2005); Physica A 349, 60 (2005)] for volume transport and attendant mass and volume velocities as well as the effects of volume transport on the Newtonian law of viscosity, compression/dilatation (bulk viscosity) phenomena, and Fourier's law of heat conduction. On the basis of study made, it is concluded that the notion of volume transport is sufficiently significant to retain in irreversible thermodynamics of fluids and fluid mechanics.
Diffusion equation and spin drag in spin-polarized transport
DEFF Research Database (Denmark)
Flensberg, Karsten; Jensen, Thomas Stibius; Mortensen, Asger
2001-01-01
We study the role of electron-electron interactions for spin-polarized transport using the Boltzmann equation, and derive a set of coupled transport equations. For spin-polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two-spin...... species. This "spin drag" effect enhances the resistivity of the system. The enhancement is stronger the lower the dimension is, and should be measurable in, for example, a two-dimensional electron gas with ferromagnetic contacts. We also include spin-flip scattering, which has two effects......: it equilibrates the spin density imbalance and, provided it has a non-s-wave component, also a current imbalance....
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation
Ren, Feng; Song, Baowei; Sukop, Michael C.; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015), 10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2016-02-01
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.
Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations
Fan, W C; Powell, J L
2002-01-01
The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.
Theory of Electron Transport in Small Semiconductor Devices Using the Pauli Master Equation
M. V. Fischetti
1998-01-01
It is argued that the Pauli master equation can be used to simulate electron transport in very small electronic devices under steady-state conditions. Written in a basis of suitable wavefunctions and with the appropriate open boundary conditions, this equation removes some of the approximations which render the Boltzmann equation unsatisfactory at small length-scales. The main problems consist in describing the interaction of the system with the reservoirs and in assessing the ...
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-01
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions.
International Nuclear Information System (INIS)
A spherical harmonics research code (DANTE) has been developed which is compatible with parallel computer architectures. DANTE provides 3-D, multi-material, deterministic, transport capabilities using an arbitrary finite element mesh. The linearized Boltzmann transport equation is solved in a second order self-adjoint form utilizing a Galerkin finite element spatial differencing scheme. The core solver utilizes a preconditioned conjugate gradient algorithm. Other distinguishing features of the code include options for discrete-ordinates and simplified spherical harmonics angular differencing, an exact Marshak boundary treatment for arbitrarily oriented boundary faces, in-line matrix construction techniques to minimize memory consumption, and an effective diffusion based preconditioner for scattering dominated problems. Algorithm efficiency is demonstrated for a massively parallel SIMD architecture (CM-5), and compatibility with MPP multiprocessor platforms or workstation clusters is anticipated
Lattice Boltzmann Simulation of Multiphase Transport in Nanostructured PEM Fuel Cells
Stiles, Christopher D.
As the fossil fuel crisis becomes more critical, it is imperative to develop renewable sources of power generation. Polymer electrolyte membrane (PEM) fuel cells are considered a viable option. However, the cost of the platinum catalyst has hindered their commercialization. PEM fuel cells with platinum loading of >0.4 mg cm2 are common. Efforts towards further reducing this loading are currently underway utilizing nanostructured electrodes. A consequence of increased platinum utilization per unit area and thinner nanostructured electrodes is flooding, which is detrimental to fuel cell performance. Flooding causes a two-fold impact on cell performance: a drop in cell voltage and a rise in parasitic pumping power to overcome the increased pressure drop, which together result in a significant reduction in system efficiency. Proper water management is therefore crucial for optimum performance of the fuel cell and also for enhancing membrane durability. The goal of this thesis is to simulate the multiphase fluid transport in the nanostructured PEMFC of H2O in air with realistic density ratios. In order to pursue this goal, the ability of the pseudopotential based multiphase lattice Boltzmann method to realistically model the coexistence of the gas and liquid phases of H2O at low temperatures is explored. This method is expanded to include a gas mixture of O2 and N 2 into the multiphase H2O systems. Beginning with the examination of the phase transition region described by the current implementation of the multiphase pseudopotential lattice Boltzmann model. Following this, a modified form of the pressure term with the use of a scalar multiplier kappa for the Peng-Robinson equation of state is thoroughly investigated. This method proves to be very effective at enabling numerically stable simulations at low temperatures with large density ratios. It is found that for decreasing values of kappa, this model leads to an increase in multiphase interface thickness and a
Robson, R E; Winkler, R; Sigeneger, F
2002-05-01
The Boltzmann equation corresponding to a general "multiterm" representation of the phase space distribution function f(r,c,t) for charged particles in a gas in an electric field was reformulated entirely in terms of spherical tensors f(l)(m) some time ago, and numerous applications, including extension to time varying and crossed electric and magnetic fields, have followed. However, these applications have, by and large, been limited to the hydrodynamic conditions that prevail in swarm experiments and the full potential of the tensor formalism has thus never been realized. This paper resumes the discussion in the context of the more general nonhydrodynamic situation. Geometries for which a simple Legendre polynomial expansion suffices to represent f are discussed briefly, but the emphasis is upon cylindrical geometry, where such simplification does not arise. In particular, we consider an axisymmetric cylindrical column of weakly ionized plasma, and derive an infinite hierarchy of integrodifferential equations for the expansion coefficients of the phase space distribution function, valid for both electrons and ions, and for all types of binary interaction with neutral gas molecules. PMID:12059718
International Nuclear Information System (INIS)
External linking scripts between Monte Carlo transport codes and burnup codes, and complete integration of burnup capability into Monte Carlo transport codes, have been or are currently being developed. Monte Carlo linked burnup methodologies may serve as an excellent benchmark for new deterministic burnup codes used for advanced systems; however, there are some instances where deterministic methodologies break down (i.e., heavily angularly biased systems containing exotic materials without proper group structure) and Monte Carlo burn up may serve as an actual design tool. Therefore, researchers are also developing these capabilities in order to examine complex, three-dimensional exotic material systems that do not contain benchmark data. Providing a reference scheme implies being able to associate statistical errors to any neutronic value of interest like k(eff), reaction rates, fluxes, etc. Usually in Monte Carlo, standard deviations are associated with a particular value by performing different independent and identical simulations (also referred to as 'cycles', 'batches', or 'replicas'), but this is only valid if the calculation itself is not biased. And, as will be shown in this paper, there is a bias in the methodology that consists of coupling transport and depletion codes because Bateman equations are not linear functions of the fluxes or of the reaction rates (those quantities being always measured with an uncertainty). Therefore, we have to quantify and correct this bias. This will be achieved by deriving an unbiased minimum variance estimator of a matrix exponential function of a normal mean. The result is then used to propose a reference scheme to solve Boltzmann/Bateman coupled equations, thanks to Monte Carlo transport codes. Numerical tests will be performed with an ad hoc Monte Carlo code on a very simple depletion case and will be compared to the theoretical results obtained with the reference scheme. Finally, the statistical error propagation
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
On anisotropy function in crystal growth simulations using Lattice Boltzmann equation
Younsi, Amina
2016-01-01
In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous...
Hashimoto, K.; Kanki, K.; Tanaka, S.; Petrosky, T.
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed.
Energy Technology Data Exchange (ETDEWEB)
Borges, P. D., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu; Scolfaro, L., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu [Department of Physics, Texas State University, San Marcos, Texas 78666 (United States)
2014-12-14
The thermoelectric properties of indium nitride in the most stable wurtzite phase (w-InN) as a function of electron and hole concentrations and temperature were studied by solving the semiclassical Boltzmann transport equations in conjunction with ab initio electronic structure calculations, within Density Functional Theory. Based on maximally localized Wannier function basis set and the ab initio band energies, results for the Seebeck coefficient are presented and compared with available experimental data for n-type as well as p-type systems. Also, theoretical results for electric conductivity and power factor are presented. Most cases showed good agreement between the calculated properties and experimental data for w-InN unintentionally and p-type doped with magnesium. Our predictions for temperature and concentration dependences of electrical conductivity and power factor revealed a promising use of InN for intermediate and high temperature thermoelectric applications. The rigid band approach and constant scattering time approximation were utilized in the calculations.
Li, Zhihui; Wu, Junlin; Ma, Qiang; Jiang, Xinyu; Zhang, Hanxin
2014-12-01
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Tracy, C. A.; Widom, H.
1997-01-01
Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.
Energy Technology Data Exchange (ETDEWEB)
Chang, B
2004-03-22
This paper contains three analytical solutions of transport problems which can be used to test ray-effect errors in the numerical solutions of the Boltzmann Transport Equation (BTE). We derived the first two solutions and the third was shown to us by M. Prasad. Since this paper is intended to be an internal LLNL report, no attempt was made to find the original derivations of the solutions in the literature in order to cite the authors for their work.
On the transparent conducting oxide Al doped ZnO: First Principles and Boltzmann equations study
Energy Technology Data Exchange (ETDEWEB)
Slassi, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Naji, S. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Department of Physics, Faculty of Science, Ibb University, Ibb (Yemen); Benyoussef, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Hamedoun, M., E-mail: hamedoun@hotmail.com [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); El Kenz, A. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco)
2014-08-25
Highlights: • The incorporation of Al in ZnO increases the optical band edge absorption. • Incorporated Al creates shallow donor states of Al-3s around Fermi level. • Transmittance decreases in the visible and IR regions, while it increases in the UV region. • Electrical conductivity increases and reaches almost the saturation for high concentration of Al. - Abstract: We report, in this work, a theoretical study on the electronic, optical and electrical properties of pure and Al doped ZnO with different concentrations. In fact, we investigate these properties using both First Principles calculations within TB-mBJ approximation and Boltzmann equations under the constant relaxation time approximation for charge carriers. It is found out that, the calculated lattice parameters and the optical band gap of pure ZnO are close to the experimental values and in a good agreement with the other theoretical studies. It is also observed that, the incorporations of Al in ZnO increase the optical band edge absorption which leads to a blue shift and no deep impurities levels are induced in the band gap as well. More precisely, these incorporations create shallow donor states around Fermi level in the conduction band minimum from mainly Al-3s orbital. Beside this, it is found that, the transmittance is decreased in the visible and IR regions, while it is significantly improved in UV region. Finally, our calculations show that the electrical conductivity is enhanced as a result of Al doping and it reaches almost the saturation for high concentration of Al. These features make Al doped ZnO a transparent conducting electrode for optoelectronic device applications.
Chakavorty, Arghya; Li, Lin; Alexov, Emil
2016-10-30
Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson-Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔGelec ) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔGelec . We argue that the sensitivity of the absolute ΔΔGelec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM-27, AMBER-94, and OPLS-AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔGelec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔGelec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.
Telegrapher's equation for light derived from the transport equation
Hoenders, Bernhard J.; Graaff, R.
2005-01-01
Shortcomings of diffusion theory when applied to turbid media such as biological tissue makes the development of more accurate equations desirable. Several authors developed telegrapher's equations in the well known P-1 approximation. The method used in this paper is different: it is based on the asymptotic evaluation of the solutions of the equation of radiative transport with respect to place and time for all values of the albedo. Various coefficients for the telegrapher's equations were de...
A lattice-Boltzmann scheme of the Navier-Stokes equations on a 3D cuboid lattice
Min, Haoda; Peng, Cheng; Wang, Lian-Ping
2015-11-01
The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, by adding a free parameter in the definition of moments or by extending the equilibrium moments. Here we developed a lattice Boltzmann model on 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed our MRT-LBM model by matching the moment equations from the Chapman-Enskog expansion with the Navier-Stokes equations. The model guarantees correct hydrodynamics. A second-order term is added to the equilibrium moments in order to restore the isotropy of viscosity on a cuboid lattice. The form and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the viscosity can be adjusted independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model. The resulting cuboid MRT-LBM model is then validated through benchmark simulations using laminar channel flow, turbulent channel flow, and the 3D Taylor-Green vortex flow.
Homogenization of ordinary and linear transport equations
Peirone, Roberto
1996-01-01
The homogenization of first order ordinary differential equations in $\\mathbb{R}^N$ and associated linear transport equations are studied. We prove the equivalence between $G$-convergence and strong $G$-convergence for the ordinary equations. We give a sufficient condition, which is also necessary in the autonomous case, for the weak homogenization of the linear transport equations. This condition is satisfied when div$_x f=0$.
Relativistic Boltzmann theory for a plasma
International Nuclear Information System (INIS)
This thesis gives a self-contained treatment of the relativistic Boltzmann theory for a plasma. Here plasma means any mixture containing electrically charged particles. The relativistic Boltzmann equation is linearized for the case of a plasma. The Chapman-Enskog method is elaborated further for transport phenomena. Linear laws for viscous phenomena are derived. Then the collision term in the Boltzmann theory is dealt with. Using the transport equation, a kinetic theory of wave phenomena is developed and the dissipation of hydromagnetic waves in a relativistic plasma is investigated. In the final chapter, it is demonstrated how the relativistic Boltzmann theory can be applied in cosmology. In doing so, expressions are derived for the electric conductivity of the cosmological plasma in the lepton era, the plasma era and the annihilation era. (Auth.)
Ausloos, M.
2000-09-01
Recent observations have indicated that the traditional equilibrium market hypothesis (EMH; also known as Efficient Market Hypothesis) is unrealistic. It is shown here that it is the analog of a Boltzmann equation in physics, thus having some bad properties of mean-field approximations like a Gaussian distribution of price fluctuations. A kinetic theory for prices can be simply derived, considering in a first approach that market actors have all identical relaxation times, and solved within a Chapman-Enskog like formalism. In closing the set of equations, (i) an equation of state with a pressure and (ii) the equilibrium (isothermal) equation for the price (taken as the order parameter) of a stock as a function of the volume of money available are obtained.
Lattice Boltzmann equation calculation of internal, pressure-driven turbulent flow
Hammond, L A; Care, C M; Stevens, A
2002-01-01
We describe a mixing-length extension of the lattice Boltzmann approach to the simulation of an incompressible liquid in turbulent flow. The method uses a simple, adaptable, closure algorithm to bound the lattice Boltzmann fluid incorporating a law-of-the-wall. The test application, of an internal, pressure-driven and smooth duct flow, recovers correct velocity profiles for Reynolds number to 1.25 x 10 sup 5. In addition, the Reynolds number dependence of the friction factor in the smooth-wall branch of the Moody chart is correctly recovered. The method promises a straightforward extension to other curves of the Moody chart and to cylindrical pipe flow.
Premnath, Kannan N; Pattison, Martin J; Banerjee, Sanjoy
2009-02-01
In this paper, we present a framework based on the generalized lattice Boltzmann equation (GLBE) using multiple relaxation times with forcing term for eddy capturing simulation of wall-bounded turbulent flows. Due to its flexibility in using disparate relaxation times, the GLBE is well suited to maintaining numerical stability on coarser grids and in obtaining improved solution fidelity of near-wall turbulent fluctuations. The subgrid scale (SGS) turbulence effects are represented by the standard Smagorinsky eddy viscosity model, which is modified by using the van Driest wall-damping function to account for reduction of turbulent length scales near walls. In order to be able to simulate a wider class of problems, we introduce forcing terms, which can represent the effects of general nonuniform forms of forces, in the natural moment space of the GLBE. Expressions for the strain rate tensor used in the SGS model are derived in terms of the nonequilibrium moments of the GLBE to include such forcing terms, which comprise a generalization of those presented in a recent work [Yu, Comput. Fluids 35, 957 (2006)]. Variable resolutions are introduced into this extended GLBE framework through a conservative multiblock approach. The approach, whose optimized implementation is also discussed, is assessed for two canonical flow problems bounded by walls, viz., fully developed turbulent channel flow at a shear or friction Reynolds number (Re) of 183.6 based on the channel half-width and three-dimensional (3D) shear-driven flows in a cubical cavity at a Re of 12 000 based on the side length of the cavity. Comparisons of detailed computed near-wall turbulent flow structure, given in terms of various turbulence statistics, with available data, including those from direct numerical simulations (DNS) and experiments showed good agreement. The GLBE approach also exhibited markedly better stability characteristics and avoided spurious near-wall turbulent fluctuations on coarser grids
Conjugate heat and mass transfer in the lattice Boltzmann equation method.
Li, Like; Chen, Chen; Mei, Renwei; Klausner, James F
2014-04-01
An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved
Thermal Lattice Boltzmann Model for Compressible Fluid
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
We formulate a new thermal lattice Boltzmann model to simulate compressible flows with a high Mach number.The main difference from the standard lattice Boltzmann models is that the particle velocities are no longer a constant, varying with the mean velocity and internal energy. The proper heat conduction term in the energy equation is recovered by modification of the fluctuating kinetic energy transported by particles. The simulation of Couette flow is in good agreement with the analytical solutions.
Three-Dimensional Lattice Boltzmann Simulation of Liquid Water Transport in Porous Layer of PEMFC
Directory of Open Access Journals (Sweden)
Bo Han
2015-12-01
Full Text Available A three-dimensional two-phase lattice Boltzmann model (LBM is implemented and validated for qualitative study of the fundamental phenomena of liquid water transport in the porous layer of a proton exchange membrane fuel cell (PEMFC. In the present study, the three-dimensional microstructures of a porous layer are numerically reconstructed by a random generation method. The LBM simulations focus on the effects of the porous layer porosity and boundary liquid saturation on liquid water transport in porous materials. Numerical results confirm that liquid water transport is strongly affected by the microstructures in a porous layer, and the transport process prefers the large pores as its main pathway. The preferential transport phenomenon is more profound with a decreased porous layer porosity and/or boundary liquid saturation. In the transport process, the breakup of a liquid water stream can occur under certain conditions, leading to the formation of liquid droplets inside the porous layer. This phenomenon is related to the connecting bridge or neck resistance dictated by the surface tension, and happens more frequently with a smaller porous layer porosity. Results indicate that an optimized design of porous layer porosity and the combination of various pore sizes may improve both the liquid water removal and gaseous reactant transport in the porous layer of a PEMFC.
S, Savithiri; Pattamatta, Arvind; Das, Sarit K
2015-01-01
Severe contradictions exist between experimental observations and computational predictions regarding natural convective thermal transport in nanosuspensions. The approach treating nanosuspensions as homogeneous fluids in computations has been pin pointed as the major contributor to such contradictions. To fill the void, inter particle and particle fluid interactivities (slip mechanisms), in addition to effective thermophysical properties, have been incorporated within the present formulation. Through thorough scaling analysis, the dominant slip mechanisms have been identified. A Multi Component Lattice Boltzmann Model (MCLBM) approach has been proposed, wherein the suspension has been treated as a non homogeneous twin component mixture with the governing slip mechanisms incorporated. The computations based on the mathematical model can accurately predict and quantify natural convection thermal transport in nanosuspensions. The role of slip mechanisms such as Brownian diffusion, thermophoresis, drag, Saffman ...
A Discussion on Whether 15-20C Are All Skin Nuclei via Isospin-dependent Boltzmann-Langevin Equation
Institute of Scientific and Technical Information of China (English)
CHEN Yu; ZHANG Feng-Shou; SU Jun
2009-01-01
A new attempt of calculation for the total reaction cross sections (σR) has been carried out within the isospin-dependent Boltzmann-Langevin equation in the intermediate energy heavy-ion collision of isotopes of G. The σR of both stable and exotic nuclei are reproduced rather well. The incident energy and isospin dependencies of σR have been investigated. It is found that the isospin effect is comparatively remarkable at intermediate energy. It is also found that ~(15-18)C are neutron skin nuclei but for ~(19)C and ~(20)C we cannot draw a conclusion whether they have halo structures.
Onsager equations and time dependent neutron transport
International Nuclear Information System (INIS)
The diffusion of neutrons following an abrupt, localized temperature fluctuation can be conducted in the framework of Onsager-type transport equations. Considering Onsager equations as a generalized Fick's law, time-dependent particle and energy 'generalized diffusion equations' can be obtained. Aim of the present paper is to obtain the time-dependent diffusion Onsager-type equations for the diffusion of neutrons and to apply them to simple trial cases to gain a feeling for their behaviour. (author)
Hu, Kainan; Geng, Shaojuan
2016-01-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e. the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion...
Chen, Li; Kang, Qinjun; Yao, Jun; Tao, Wenquan
2014-01-01
Porous structures of shales are reconstructed based on scanning electron microscopy (SEM) images of shale samples from Sichuan Basin, China. Characterization analyzes of the nanoscale reconstructed shales are performed, including porosity, pore size distribution, specific surface area and pore connectivity. The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) fluid flow model and single-relaxation-time (SRT) LBM diffusion model are adopted to simulate the fluid flow and Knudsen diffusion process within the reconstructed shales, respectively. Tortuosity, intrinsic permeability and effective Knudsen diffusivity are numerically predicted. The tortuosity is much higher than that commonly employed in Bruggeman equation. Correction of the intrinsic permeability by taking into consideration the contribution of Knudsen diffusion, which leads to the apparent permeability, is performed. The correction factor under different Knudsen number and pressure are estimated and compared with existing corrections re...
Transport Properties of the Universal Quantum Equation
Institute of Scientific and Technical Information of China (English)
A.I.Arbab
2012-01-01
The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles.This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime.A new transformation that sends the Schr(o)dinger equation with a potential energy V =-1/2mc2 to Dirac's equation is proposed.The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations.Furthermore,a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived.This equation is found to govern a Klein Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.
Ying, Jinyong
2016-01-01
The size-modified Poisson-Boltzmann equation (SMPBE) is one important variant of the popular dielectric model, the Poisson-Boltzmann equation (PBE), to reflect ionic size effects in the prediction of electrostatics for a biomolecule in an ionic solvent. In this paper, a new SMPBE hybrid solver is developed using a solution decomposition, the Schwartz's overlapped domain decomposition, finite element, and finite difference. It is then programmed as a software package in C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS project. This software package is well validated on a Born ball model with analytical solution and a dipole model with a known physical properties. Numerical results on six proteins with different net charges demonstrate its high performance. Finally, this new SMPBE hybrid solver is shown to be numerically stable and convergent in the calculation of electrostatic solvation free energy for 216 biomolecules and binding free energy for a DNA-drug com...
Transport in the spatially tempered, fractional Fokker-Planck equation
Energy Technology Data Exchange (ETDEWEB)
Kullberg, A. [University of California, Los Angeles; Del-Castillo-Negrete, Diego B [ORNL
2012-01-01
A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, lambda, and on the order of the fractional derivative in space, alpha. An expansion of the TFD operator for large lambda is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit lambda -> infinity, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (lambda > 0) truncations, and alpha not equal 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit lambda -> infinity, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for alpha not equal 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any lambda > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with lambda, as J similar to lambda(-zeta), for alpha >= 1.75. However, for alpha <= 1.5, the steady-state current decays exponentially with lambda, as J similar to e(-xi lambda). In the presence of an asymmetry in the TFD operator, the tempering can lead
Transport in the spatially tempered, fractional Fokker-Planck equation
Kullberg, A.; del-Castillo-Negrete, D.
2012-06-01
A study of truncated Lévy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, λ, and on the order of the fractional derivative in space, α. An expansion of the TFD operator for large λ is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit λ → ∞, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (λ > 0) truncations, and α ≠ 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit λ → ∞, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for α ≠ 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any λ > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with λ, as J ˜ λ-ζ, for α ⩾ 1.75. However, for α ⩽ 1.5, the steady-state current decays exponentially with λ, as J ˜ e-ξλ. In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the
Lattice Boltzmann technique for heat transport phenomena coupled with melting process
Ibrahem, A. M.; El-Amin, M. F.; Mohammadein, A. A.; Gorla, Rama Subba Reddy
2016-04-01
In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar-Gross-Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.
Lattice Boltzmann Method for Diffusion-Reaction-Transport Processes in Heterogeneous Porous Media
Institute of Scientific and Technical Information of China (English)
XU You-Sheng; ZHONG Yi-Jun; HUANG Guo-Xiang
2004-01-01
Based on the lattice Boltzmann method and general theory of fluids flowing in porous media, a numerical model is presented for the diffusion-reaction-transport (DRT) processes in porous media. As a test, we simulate a DRT process in a two-dimensional horizontal heterogeneous porous medium. The influence of gravitation in this case can be neglected, and the DRT process can be described by a strongly heterogeneous diagnostic test strip or a thin confined piece of soil with stochastically distributing property in horizontal directions. The results obtained for the relations between reduced fluid saturation S, concentration c1, and concentration c2 are shown by using the visualization computing technique. The computational efficiency and stability of the model are satisfactory.
Lahiri, T.; Pal Majumder, T.; Ghosh, N. K.
2014-07-01
Commercialization of ferroelectric liquid crystal displays (FLCDs) suffers from mechanical and electro-convective instabilities. Impurity ions play a pivotal role in the latter case, and therefore we developed a mean-field type model to understand the complex role of space charges, particularly ions in a ferroelectric liquid crystal. Considering an effective ion-chirality relation, we obtained a modified Poisson-Boltzmann equation for ions dissolved into a chiral solvent like the ferroelectric smectic phase. A nonuniform director profile induced by the mean electrostatic potential of the ions is then calculated by solving an Euler-Lagrange equation for a helically twisted smectic state. A combination of effects resulting from molecular chirality and an electrostatically driven twist created by the ions seems to produce this nonuniform fluctuation in the director orientation. Finally, both theoretical and experimental points of view are presented on the prediction of this mean-field model.
Anomalous transport equations in toroidal plasmas
International Nuclear Information System (INIS)
Reduced transport equations for a toroidal plasma with fluctuations are derived. These equations include the effects of both anomalous and standard neoclassical transport, and allow clarification of the structure of convective fluxes caused by electrostatic and magnetic fluctuations. Special attention is paid to the combined effects of fluctuations and toroidicity on the transport. The formulation retains the effects of a magnetic field inhomogeneity on the anomalous transport. It is shown that phase space diffusion caused by the gradient in the equilibrium magnetic field appears as a pinch flux in the real space
Mixing and Transport in the Small Intestine: A Lattice-Boltzmann Model
Banco, Gino; Brasseur, James; Wang, Yanxing; Aliani, Amit; Webb, Andrew
2007-11-01
The two primary functions of the small intestine are absorption of nutrients into the blood stream and transport of material along the gut for eventual evacuation. The primary transport mechanism is peristalsis. The time scales for absorption, however, rely on mixing and transport of molecules between the bulk flow and epithelial surface. Two basic motions contribute to mixing: peristalsis and repetitive segmental contraction of short segments of the gut. In this study we evaluate the relative roles of peristalsis vs. segmental contraction on the degree of mixing and time scales of nutrient transport to the epithelium using a two-dimensional model of flow and mixing in the small intestine. The model uses the lattice-Boltzmann framework with second-order moving boundary conditions and passive scalar (Sc = 10). Segmental and peristaltic contractions were parameterized using magnetic resonance imaging data from rat models. The Reynolds numbers (1.9), segment lengths (33 mm), max radii (2.75 mm) and occlusion ratios (0.33) were matched for direct comparison. Mixing is quantified by the rate of dispersion of scalar from an initial concentration in the center of the segment. We find that radial mixing is more rapid with segmental than peristaltic motion, that radial dispersion is much more rapid than axial, and that axial is comparable between the motions.
Accurate deterministic solutions for the classic Boltzmann shock profile
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Chen, Li; Kang, Qinjun; Holby, Edward F; Tao, Wen-Quan
2014-01-01
High-resolution porous structures of catalyst layer (CL) with multicomponent in proton exchange membrane fuel cells are reconstructed using a reconstruction method called quartet structure generation set. Characterization analyses of nanoscale structures are implemented including pore size distribution, specific area and phase connectivity. Pore-scale simulation methods based on the lattice Boltzmann method are developed and used to predict the macroscopic transport properties including effective diffusivity and proton conductivity. Nonuniform distributions of ionomer in CL generates more tortuous pathway for reactant transport and greatly reduces the effective diffusivity. Tortuosity of CL is much higher than conventional Bruggeman equation adopted. Knudsen diffusion plays a significant role in oxygen diffusion and significantly reduces the effective diffusivity. Reactive transport inside the CL is also investigated. Although the reactive surface area of non-precious metal catalyst (NPMC) CL is much higher t...
Dual Diagonalization of Reactive Transport Equations
Yeh, G.; Tsai, C.
2013-12-01
One solves a system of species transport equations in the primitive approach to reactive transport modeling. This approach is not able to decouple equilibrium reaction rates from species concentrations. This problem has been overcome with the approach to diagonalizing the reaction matrix since mid 1990's, which yields the same number of transport equations for reaction-extents. In the diagonalization approach, first, a subset of reaction- extent transport equations is solved for concentrations of components and kinetic-variables. Then, the component, kinetic-variable, and mass action equations are solved for all species concentrations. Finally, the equilibrium reaction rates are posterior computed. The difficulty in this approach is that the solution of species concentrations in the second step is a stiff problem when the concentrations of master species are small compared to those of equilibrium species. To overcome the problem of stiffness, we propose a dual diagonalization approach. Here, a second diagonalization is performed on the decomposed unit matrix to yield species concentrations, each as a linear function of reaction extents. In this dual diagonalization approach, four steps are needed to complete the modeling. First, component and kinetic-variable transport equations are solved for the concentrations of components (a subset of reaction-extents) and kinetic-variables (another subset of reaction-extents). Second, the set of mass action equations written in terms of reaction extents are solved for equilibrium-variables (yet another subset of reaction-extents). Third, species concentrations are posterior obtained by solving the set of linear equations defining reaction-extents. Fourth, equilibrium rates are posterior calculated with transport equations for equilibrium-variables. Several example problems will be used to demonstrate the efficiency of this approach. Keywords: Reactive Transport, Reaction-Extent, Component, Kinetic-Variable, Equilibrium
Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Pontrelli, G.; Evans, P. C.
2016-08-01
We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013), 10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.
Energy-Dependent Boltzmann Equation with Fission and Slowing-Down Kernels
International Nuclear Information System (INIS)
This paper presents a study of the energy-dependent neutron transport equation, using Case's method of singular eigenfunctions and considering a continuous energy variable (rather than a multigroup scheme). Both fission and slowing-down kernels are included in the analysis. Under the assumption of simple cross-sections laws, and plane symmetry, a completeness theorem and the Green's function are found for-the infinite medium, for both isotropic and anisotropic scattering, using rather general assumptions as to the slowing-down kernels (including convolution kernels) and, only in the anisotropic case, the generalized Greuling-Goertzel kernel. The crux of the completeness theorem is the inversion and analysis of the spectral properties of a continuous operator which acts upon the energy variable, and depends parametrically upon a complex variable z (with analyticity in some cut complex plane). For half-space problems, the Wiener-Hopf factorization of such an operator is a remarkably difficult problem. However, it can be performed if, for the slowing-down kernel, the Greuling-Goertzel approximation generalized to all its anisotropic components is used, in which case the Wiener-Hopf factorization gives another convolution operator. In this approximation the Milne problem is solved, and a study is made of the extrapolation length. There is a discussion of the difficulties introduced by fission kernels, with emphasis on the coexistence of space-energy separable modes with slowing-down transient modes. (author)
Xiong, Yuan
2014-04-28
Spurious current emerging in the vicinity of phase interfaces is a well-known disadvantage of the lattice Boltzmann equation (LBE) for two-phase flows. Previous analysis shows that this unphysical phenomenon comes from the force imbalance at discrete level inherited in LBE (Guo et al 2011 Phys. Rev. E 83 036707). Based on the analysis of the LBE free of checkerboard effects, in this work we further show that the force imbalance is caused by the different discretization stencils: the implicit one from the streaming process and the explicit one from the discretization of the force term. Particularly, the total contribution includes two parts, one from the difference between the intrinsically discretized density (or ideal gas pressure) gradient and the explicit ones in the force term, and the other from the explicit discretized chemical potential gradients in the intrinsically discretized force term. The former contribution is a special feature of LBE which was not realized previously.
Dyatko, Nikolay
2015-01-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This bistability effect - in which electron-electron (Coulomb) collisions play an essential role - is analyzed here with a Boltzmann equation approach and with a first principles particle simulation method. The latter is based on a combination of a molecular dynamics technique that accounts for the many-body interaction within the electron gas and a Monte Carlo treatment of the collisions between electrons and the background gas atoms. The good agreement found between the results of the two techniques confirms the existence of the two different stable solutions for the EEDF under swarm conditions at low electric fields.
Allaire, Gregoire; Dufreche, Jean-Francois; Mikelic, Andro; Piatnitski, Andrey
2013-01-01
This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality...
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The Boltzmann simplified velocity distribution function equation describing the gas transfer phenomena from various flow regimes will be explored and solved numerically in this study. The discrete velocity ordinate method of the gas kinetic theory is studied and applied to simulate the complex multi-scale flows. Based on the uncoupling technique on molecular movement and colliding in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by extending and applying the unsteady time-splitting method from computational fluid dynamics. The Gauss-type discrete velocity numerical quadrature technique for different Mach number flows is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established to study the three-dimensional complex flows from rarefied transition to continuum regimes. The parallel strategy adapted to the gas-kinetic numerical algorithm is investigated by analyzing the inner parallel degree of the algorithm, and then the HPF parallel processing program is developed. To test the reliability of the present gas-kinetic numerical method, the three-dimensional complex flows around sphere and spacecraft shape with various Knudsen numbers are simulated by HPF parallel computing. The computational results are found in high resolution of the flow fields and good agreement with the theoretical and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the Boltzmann model equation.
Adaptive integral equation methods in transport theory
International Nuclear Information System (INIS)
In this paper, an adaptive multilevel algorithm for integral equations is described that has been developed with the Chandrasekhar H equation and its generalizations in mind. The algorithm maintains good performance when the Frechet derivative of the nonlinear map is singular at the solution, as happens in radiative transfer with conservative scattering and in critical neutron transport. Numerical examples that demonstrate the algorithm's effectiveness are presented
Ender, I A; Flegontova, E Yu; Gerasimenko, A B
2016-01-01
An algorithm for sequential calculation of non-isotropic matrix elements of the collision integral which are necessary for the solution of the non-linear Boltzmann equation by moment method is proposed. Isotropic matrix elements that we believe are known, are starting ones. The procedure is valid for any interaction law and any mass ratio of the colliding particles.
Basu, Prasad
2011-01-01
Following the original approach of Maxwell-Boltzmann(MB), we derive a 4-velocity distribution function for the relativistic ideal gas. This distribution function perfectly reduces to original MB distribution in the non-relativistic limit. We express the relativistic equation of state(EOS), $\\rho-\\rho_0=(\\gamma-1)^{-1}p$,\\ in the two equations: $\\rho=\\rho_0 f(\\lambda)$,\\ and $p=\\rho_0 g(\\lambda)$, where $\\lambda$\\ is a parameter related to the kinetic energy, hence the temperature, of the gas. In the both extreme limits, they give correct EOS:\\ $\\rho=3p$\\ in the ultra-relativistic, and\\ $\\rho-\\rho_0=3/2p$ in the non-relativistic regime. Using these equations the adiabatic index $\\gamma$ (=$\\frac{c_p}{c_v}$) and the sound speed $a_s$ are calculated as a function of $\\lambda$. They also satisfy the inequalities: $4/3 \\le \\gamma \\le 5/3$ and $a_s \\le \\frac{1}{\\sqrt{3}}$ perfectly.
Lattice Boltzmann pore-scale model for multi-component reactive transport
Kang, Qinjun; Lichtner, Peter; Zhang, Dongxiao; Tsimpanogiannis, Ioannis
2004-11-01
A better understanding of multi-component flow and reaction in natural and man-made porous media is critical to a wide range of fields, including hydrology (groundwater quality), fossil energy (oil, gas, coalbed methane, clathrates), economic geology (mineral processing and development), geologic carbon sequestration (hydrodynamic and mineral trapping of carbon), and materials manufacturing and degradation (polymer composites, concrete, building materials). This problem is notoriously difficult because it usually involves multiple processes (convection, diffusion, and chemical reaction) and complex geometries and boundaries. In this work, we present a multi-component lattice Boltzmann model for simulating reactive transport in porous media at the pore scale. This model takes into account convection, diffusion, homogeneous reactions among multiple aqueous species, heterogeneous reactions between the aqueous solution and the minerals, as well as the resulting geometrical changes in pore space. Homogeneous reactions are described through local equilibrium mass action relations. Mineral reactions are treated kinetically through boundary conditions at the surface. We have applied this model to a hypothetical two-component system in a synthetically constructed medium, and analyzed the effects of convection, diffusion, reaction rate constants, and chemical compositions on mineral alteration of the porous medium.
Aspheric surface testing by irradiance transport equation
Shomali, Ramin; Darudi, Ahmad; Nasiri, Sadollah; Asgharsharghi Bonab, Armir
2010-10-01
In this paper a method for aspheric surface testing is presented. The method is based on solving the Irradiance Transport Equation (ITE).The accuracy of ITE normally depends on the amount of the pick to valley of the phase distribution. This subject is investigated by a simulation procedure.
Boltzmann-Equation Based Derivation of Balance Laws in Irreversible Thermodynamics
Hong, Liu; Yang, Zaibao; Zhu, Yi; Yong, Wen-An
2014-01-01
In this paper we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, separating entropy fluxes and production rates properly, and determining a dissipation matrix. Our approach takes the advantage of both EIT and GENERIC form...
Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
Glangetas, Leo; Li, Hao-Guang; Xu, Chao-Jiang
2014-01-01
Accepted to publish by "Kinetic and Related Models" In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing properties is same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of this fractional is exactly the singular index of non-cutoff collisional ker...
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.
Generalized Poisson-Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects
Institute of Scientific and Technical Information of China (English)
刘新敏; 李航; 李睿; 田锐; 许晨阳
2012-01-01
Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 〈 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration 〉 0.15 tool/1 in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
Shahmardan, M. M.; Sedaghat, M. H.; Norouzi, M.; Nazari, M.
2015-12-01
Numerical simulation based on immersed boundary-lattice Boltzmann method has been employed to study 2D muco-ciliary transport problem. The periciliary liquid (PCL) and mucus layers in this study are considered as the Newtonian and viscoelastic fluid respectively. An Oldroyd-B model is used as the constitutive equations of mucus layer. To simulate accurate effects of the cilia and PCL-mucus interface on the fluid, immersed boundary method is used. Numerical simulations have been performed to investigate the effects of mucus depth on the muco-ciliary clearance at various values of cilia beat frequencies. Our results show that, by increasing mucus depth, which results from air pollution and smoking, mean mucus velocity decreases. But it can be completely modified by increasing cilia beat frequency and the cilia beat frequency has great effect on the muco-ciliary clearance.
Deng, Y. K.; Xiao, D. M.
2012-02-01
The present paper concerns itself with the insulation characteristics of c-C4F8/N2 gas mixtures and studies the possibility of applying in the gas insulation of power equipments. We aim to use the theoretical framework of the Boltzmann equation to calculate the density-normalized effective ionization coefficients (α-ƞ)/N and transport parameters of c-C4F8/N2 gas mixtures for E/N values from 180 to 550 Td (1 Td = 10-17 V cm2) in the condition of steady-state Townsend (SST) experiment. From the variation curve of (α-ƞ)/N with the c-C4F8 mixture ratio k, the limiting field strength (E/N)lim of the gas mixtures at different gas content is determined. In order to confirm the validity of the results obtained, comparisons with Monte Carlo simulation and experimental data have been performed. It is found that the insulation properties of c-C4F8 and N2 gas mixtures are much better than those of SF6 and N2 mixtures for applying in the high voltage apparatus as an insulation medium, especially if we take the global warming potential into account.
Monte Carlo variance reduction approaches for non-Boltzmann tallies
International Nuclear Information System (INIS)
Quantities that depend on the collective effects of groups of particles cannot be obtained from the standard Boltzmann transport equation. Monte Carlo estimates of these quantities are called non-Boltzmann tallies and have become increasingly important recently. Standard Monte Carlo variance reduction techniques were designed for tallies based on individual particles rather than groups of particles. Experience with non-Boltzmann tallies and analog Monte Carlo has demonstrated the severe limitations of analog Monte Carlo for many non-Boltzmann tallies. In fact, many calculations absolutely require variance reduction methods to achieve practical computation times. Three different approaches to variance reduction for non-Boltzmann tallies are described and shown to be unbiased. The advantages and disadvantages of each of the approaches are discussed
Wei, Linsheng; Xu, Min; Yuan, Dingkun; Zhang, Yafang; Hu, Zhaoji; Tan, Zhihong
2014-10-01
The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10-17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process.
International Nuclear Information System (INIS)
The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10−17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process. (low temperature plasma)
Pdf - Transport equations for chemically reacting flows
Kollmann, W.
1989-01-01
The closure problem for the transport equations for pdf and the characteristic functions of turbulent, chemically reacting flows is addressed. The properties of the linear and closed equations for the characteristic functional for Eulerian and Lagrangian variables are established, and the closure problem for the finite-dimensional case is discussed for pdf and characteristic functions. It is shown that the closure for the scalar dissipation term in the pdf equation developed by Dopazo (1979) and Kollmann et al. (1982) results in a single integral, in contrast to the pdf, where double integration is required. Some recent results using pdf methods obtained for turbulent flows with combustion, including effects of chemical nonequilibrium, are discussed.
Illg, Christian; Haag, Michael; Teeny, Nicolas; Wirth, Jens; Fähnle, Manfred
2016-03-01
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi's golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
Sman, van der R.G.M.
2014-01-01
In this paper we present a novel numerical scheme for simulating the one-dimensional deformation of hydrogel material due to drying or rehydration. The scheme is based on the versatile Lattice Boltzmann method, which has been extended such that the computational grid (lattice) deforms due to shrinka
Joshi, Abhijit; Caglan Kumbur, E.; Sun, Ying
2010-11-01
The Vanadium Redox Battery (VRB) promises to be an attractive option for storing electrical energy from renewable energy sources and delivering the stored energy to the grid whenever it is required. In this work, a novel methodology is proposed for modeling the transport mechanisms in the VRB including the flow of electrolyte, chemical species transport through the electrolyte and electrochemical reactions at active sites. The detailed geometry of the electrode is obtained using X-ray computed tomography (XCT) and this geometry is characterized to calculate porosity, pore-size distribution, connectivity and the active surface area. The processed XCT data is then used as geometry input for modeling transport processes in the VRB. The flow of electrolyte through the pore-space within the electrodes and the transport of ionic species in these pores is modelled using the lattice Boltzmann method (LBM). An electrochemical model using the Butler-Volmer equations is used to provide species flux boundary conditions at the surface of the carbon fibers and to provide the necessary coupling to the local concentration of these species present in the pore space. Finally, model predictions for the steady-state discharge of the VRB are then compared with results reported in the literature.
International Nuclear Information System (INIS)
An extensive discussion of the analytical solution for the Poisson-Boltzmann equation in cylindrical symmetry for strong polyelectrolytes in the cell model is presented. The reduced mean electrostatic potential μ at finite dilutions is discussed in terms of its dependence on the polyelectrolyte equivalent concentration Ce, its charge density parameter ξ, and the distance of closest approach a of the counterions to the polyion. It is shown that in the limit a → 0 counterion condensation is expected. For more realistic nonzero values of a, the reduced potential μ at a given relative position r/R in the cell with radius R is practically independent of the linear charge density for ξ > 2, but its value depends on the product a2Ce. The value μ(a) of the reduced potential near the surface of the polyion is ξ-dependent, however, under the same conditions. A large fraction of all the counterions in the cell accumulate, on the average, in the neighborhood of the polyion, this fraction being larger the higher ξ is and the lower the product a2Ce is. The fraction of ions accumulated between the polyion surface at a and a distance from the polyion axis equal to the screening length 1/χ is high, reaching values exceeding 80% and being higher the smaller a2Ce is. This fraction of counterions (the open-quotes associatedclose quotes counterions) occupies a smaller part of the total cell volume than the counterions situated between 1/χ and R, which are characterized by a relatively low electrostatic interaction energy with the polyion, μ < 1 (the open-quotes freeclose quotescounterions). 22 refs., 11 figs
Mishonov, Todor M.; Maneva, Yana G.
2006-01-01
The differential conductivity for the out-of-plane transport in layered cuprates is calculated for Lawrence-Doniach model in the framework of time-dependent Ginzburg-Landau (TDGL) theory. The TDGL equation for the superconducting order parameter is solved in the presence of Langevin external noise, describing the birth of fluctuation Cooper pairs. The TDGL correlator of the superconducting order parameter is calculated in momentum representation and it is shown that the so defined number of p...
Anomalous Fractional Diffusion Equation for Transport Phenomena
Institute of Scientific and Technical Information of China (English)
QiuhuaZENG; HouqiangLI; 等
1999-01-01
We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.
Lattice Boltzmann Large Eddy Simulation Model of MHD
Flint, Christopher
2016-01-01
The work of Ansumali \\textit{et al.}\\cite{Ansumali} is extended to Two Dimensional Magnetohydrodynamic (MHD) turbulence in which energy is cascaded to small spatial scales and thus requires subgrid modeling. Applying large eddy simulation (LES) modeling of the macroscopic fluid equations results in the need to apply ad-hoc closure schemes. LES is applied to a suitable mesoscopic lattice Boltzmann representation from which one can recover the MHD equations in the long wavelength, long time scale Chapman-Enskog limit (i.e., the Knudsen limit). Thus on first performing filter width expansions on the lattice Boltzmann equations followed by the standard small Knudsen expansion on the filtered lattice Boltzmann system results in a closed set of MHD turbulence equations provided we enforce the physical constraint that the subgrid effects first enter the dynamics at the transport time scales. In particular, a multi-time relaxation collision operator is considered for the density distribution function and a single rel...
Institute of Scientific and Technical Information of China (English)
DOU Zhi; ZHOU Zhi-fang; HUANG Yong; WU Wei
2012-01-01
In this article,the Non-Aqueous Phase Liquid (NAPL) transport in the single filled fracture was studied with the Shan-Chen multi-component multiphase Lattice Boltzmann Method (LBM) with special consideration of wettability effects.With the help of the model,the contact angle of the non-wetting phase and wetting phase interface at a solid wall could be adjusted.By considering a set of appropriate boundary conditions,the fractured conductivity was investigated in condition that the NAPL blocks the channels in the single filled fracture.In order to study the wettability effects on the NAPL transport,a constant driving force was introduced in the Shan-Chen multi-component multiphase LBM.Flow regimes with different wettabilities were discussed.Simulated results show that the LBM is a very instrumental method for simulating and studying the immiscible multiphase flow problems in single filled fracture.
Cao, Shanshan; Qin, Guang-You; Wang, Xin-Nian
2016-01-01
A Linearized Boltzmann Transport (LBT) model coupled with hydrodynamical background is established to describe the evolution of jet shower partons and medium excitations in high energy heavy-ion collisions. We extend the LBT model to include both elastic and inelastic processes for light and heavy partons in the quark-gluon plasma. A hybrid model of fragmentation and coalescence is developed for the hadronization of heavy quarks. Within this framework, we investigate how heavy flavor observables depend on various ingredients, such as different energy loss and hadronization mechanisms, the momentum and temperature dependences of the transport coefficients, and the radial flow of the expanding fireball. Our model calculations show good descriptions of $D$ meson suppression and elliptic flow observed at the LHC and RHIC. The prediction for the Pb-Pb collisions at $\\sqrt{s_\\mathrm{NN}}$=5.02 TeV is provided.
Asymptotic Analysis of Transport Equation in Annulus
Wu, Lei; Yang, Xiongfeng; Guo, Yan
2016-09-01
We consider the diffusive limit of a steady neutron transport equation with one-speed velocity in a two-dimensional annulus. A classical theorem in Bensoussan et al. (Publ Res Inst Math Sci 15:53-157, 1979) states that the solution can be approximated in L^{∞} by the sum of the interior solution and Knudsen layer derived from Milne problem. However, this result was disproved in Wu and Guo (Commun Math Phys 336:1473-1553, 2015) in a plate via a different boundary layer expansion with geometric correction. In this paper, we established the diffusive limit and provide a counterexample to Bensoussan et al. (1979) in non-convex domains.
Energy Technology Data Exchange (ETDEWEB)
Mikell, Justin K. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Klopp, Ann H. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Gonzalez, Graciela M.N. [Department of Biostatistics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Kisling, Kelly D. [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Price, Michael J. [Department of Physics and Astronomy, Louisiana State University and Agricultural and Mechanical College, Baton Rouge, Louisiana, and Mary Bird Perkins Cancer Center, Baton Rouge, Louisiana (United States); Berner, Paula A. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Eifel, Patricia J. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Mourtada, Firas, E-mail: fmourtad@christianacare.org [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Experimental Diagnostic Imaging, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Radiation Oncology, Helen F. Graham Cancer Center, Newark, Delaware (United States)
2012-07-01
Purpose: To investigate the dosimetric impact of the heterogeneity dose calculation Acuros (Transpire Inc., Gig Harbor, WA), a grid-based Boltzmann equation solver (GBBS), for brachytherapy in a cohort of cervical cancer patients. Methods and Materials: The impact of heterogeneities was retrospectively assessed in treatment plans for 26 patients who had previously received {sup 192}Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance-compatible tandems and unshielded colpostats. The GBBS models sources, patient boundaries, applicators, and tissue heterogeneities. Multiple GBBS calculations were performed with and without solid model applicator, with and without overriding the patient contour to 1 g/cm{sup 3} muscle, and with and without overriding contrast materials to muscle or 2.25 g/cm{sup 3} bone. Impact of source and boundary modeling, applicator, tissue heterogeneities, and sensitivity of CT-to-material mapping of contrast were derived from the multiple calculations. American Association of Physicists in Medicine Task Group 43 (TG-43) guidelines and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, International Commission on Radiation Units and Measurements (ICRU) report 38 rectal and bladder points, three and nine o'clock, and {sub D2cm3} to the bladder, rectum, and sigmoid. Results: Points A and B, D{sub 2} cm{sup 3} bladder, ICRU bladder, and three and nine o'clock were within 5% of TG-43 for all GBBS calculations. The source and boundary and applicator account for most of the differences between the GBBS and TG-43 guidelines. The D{sub 2cm3} rectum (n = 3), D{sub 2cm3} sigmoid (n = 1), and ICRU rectum (n = 6) had differences of >5% from TG-43 for the worst case incorrect mapping of contrast to bone. Clinical dosimetric parameters were within 5% of TG-43 when rectal and balloon contrast were mapped to bone and radiopaque packing was not overridden
International Nuclear Information System (INIS)
Purpose: To investigate the dosimetric impact of the heterogeneity dose calculation Acuros (Transpire Inc., Gig Harbor, WA), a grid-based Boltzmann equation solver (GBBS), for brachytherapy in a cohort of cervical cancer patients. Methods and Materials: The impact of heterogeneities was retrospectively assessed in treatment plans for 26 patients who had previously received 192Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance-compatible tandems and unshielded colpostats. The GBBS models sources, patient boundaries, applicators, and tissue heterogeneities. Multiple GBBS calculations were performed with and without solid model applicator, with and without overriding the patient contour to 1 g/cm3 muscle, and with and without overriding contrast materials to muscle or 2.25 g/cm3 bone. Impact of source and boundary modeling, applicator, tissue heterogeneities, and sensitivity of CT-to-material mapping of contrast were derived from the multiple calculations. American Association of Physicists in Medicine Task Group 43 (TG-43) guidelines and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, International Commission on Radiation Units and Measurements (ICRU) report 38 rectal and bladder points, three and nine o’clock, and D2cm3 to the bladder, rectum, and sigmoid. Results: Points A and B, D2 cm3 bladder, ICRU bladder, and three and nine o’clock were within 5% of TG-43 for all GBBS calculations. The source and boundary and applicator account for most of the differences between the GBBS and TG-43 guidelines. The D2cm3 rectum (n = 3), D2cm3 sigmoid (n = 1), and ICRU rectum (n = 6) had differences of >5% from TG-43 for the worst case incorrect mapping of contrast to bone. Clinical dosimetric parameters were within 5% of TG-43 when rectal and balloon contrast were mapped to bone and radiopaque packing was not overridden. Conclusions: The GBBS has minimal impact on clinical
Blaizot, Jean-Paul; McLerran, Larry
2013-01-01
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems rea...
André, Raíla
2014-01-01
In this work we analyze the dynamics of collisionless self-gravitating systems described by the f(R)-gravity and Boltzmann equation in the weak field approximation, focusing on the Jeans instability for theses systems. The field equations in this approximation were obtained within the Palatini formalism. Through the solution of coupled equations we achieved the collapse criterion for infinite homogeneous fluid and stellar systems, which is given by a dispersion relation. This result is compared with the results of the standard case and the case for f(R)-gravity in metric formalism, in order to see the difference among them. The limit of instability varies according to which theory of gravity is adopted.
A Multiscale Lattice-Boltzmann Model of Macro-to-Micro Scale Transport Relevant to Gut Function
Wang, Yanxing; Brasseur, James; Banco, Gino
2007-11-01
Nutrient and pharmaceutical absorption in the small intestine involve coupled multiscale transport and mixing processes that span several orders of magnitude. We hypothesize that muscle-induced villi motions generate and control a ``micro-mixing layer'' that couples with macro-scale mixing to enhance molecular transport to and from the epithelium. In this work we developed a 2-D numerical method based on a multigrid strategy within the lattice-Boltzmann framework. We model a macro-scale cavity flow with microscale finger-like villi in pendular motion on the lower surface and evaluate the coupling between macro and micro-scale fluid motions, scalar mixing, and uptake of passive scalar at the villi surface. Preliminary results show that the moving villi can be effective mixers at the micro scale, especially when groups of villi move in a coordinated, but out-of-phase fashion. A time-evolving series of flow recirculation eddies are generated within a micro mixing layer that increase transport of passive scalar from the macro eddy to the surface by advection. Flow parameters such as frequency of pendular motion, spacing between villi and villi grouping, have strong influences on the behaviors of the micro-mixing layer and the efficiency of scalar transport. An extensive analysis is in process to quantify correlation between scalar mixing and flux, details of villi motion, and induced flow patterns.
Noronha, Jorge
2015-01-01
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime $\\mathrm{AdS}_{2}\\otimes \\mathrm{S}_{2}$. We further derive explicit analytic expressions for the momentum dependence of the single particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not match the equilibrium form. The non-equilibrium contribution to the entropy density is shown to be due to higher order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Thus, in this system the slowly moving hydrodynamic d...
Ludwig Boltzmann: Atomic genius
International Nuclear Information System (INIS)
On the centenary of the death of Ludwig Boltzmann, Carlo Cercignani examines the immense contributions of the man who pioneered our understanding of the atomic nature of matter. The man who first gave a convincing explanation of the irreversibility of the macroscopic world and the symmetry of the laws of physics was the Austrian physicist Ludwig Boltzmann, who tragically committed suicide 100 years ago this month. One of the key figures in the development of the atomic theory of matter, Boltzmann's fame will be forever linked to two fundamental contributions to science. The first was his interpretation of 'entropy' as a mathematically well-defined measure of the disorder of atoms. The second was his derivation of what is now known as the Boltzmann equation, which describes the statistical properties of a gas as made up of molecules. The equation, which described for the first time how a probability can evolve with time, allowed Boltzmann to explain why macroscopic phenomena are irreversible. The key point is that while microscopic objects like atoms can behave reversibly, we never see broken coffee cups reforming because it would involve a long series of highly improbable interactions - and not because it is forbidden by the laws of physics. (U.K.)
A stochastic method of solution of the Parker transport equation
Wawrzynczak, A; Gil, A
2015-01-01
We present the stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Based on the solution of the Parker transport equation we developed models of the short-time variation of the GCR intensity, i.e. the Forbush decrease (Fd) and the 27-day variation of the GCR intensity. Parker transport equation being the Fokker-Planck type equation delineates non-stationary transport of charged particles in the turbulent medium. The presented approach of the numerical solution is grounded on solving of the set of equivalent stochastic differential equations (SDEs). We demonstrate the method of deriving from Parker transport equation the corresponding SDEs in the heliocentric spherical coordinate system for the backward approach. Features indicative the preeminence of the backward approach over the forward is stressed. We compare the outcomes of the stochastic model of the Fd and 27-day variation of the GCR intensity with our former models established by the finite difference method. Both ...
Mendes, Albert C R; Abreu, Everton M C; Neto, Jorge Ananias
2016-01-01
In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac's constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition was obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev-Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan-Boltzmann type law was obtained.
Solution to the transport equation for electron backscattering on massive materials
International Nuclear Information System (INIS)
The scattering of a monoenergetic electron beam (Albedo problem) vertical to the plane of the boundary layer of a massive material is treated as transportproblem using the Boltzmann Transport Equation (BTE). The strong anistropic to 1 standardized differential effective cross section (DEC) of the single scattering of the electrons enters into the integral term. A complete, orthogonal set of solution units of the BTE can be given by a new type of approximation of the DEC in a generalization of a solution method from CASE 1960. By continuously changing an anisotropy parameter, the region of strong anisotropic and isotropic single scattering is simultaneously determined, the case results are taken as special case. (orig./RW)
Discontinuous Galerkin for the Radiative Transport Equation
Guermond, Jean-Luc
2013-10-11
This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.
Energy Technology Data Exchange (ETDEWEB)
Urbatsch, Todd James [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-06-15
We present an overview of radiation transport, covering terminology, blackbody raditation, opacities, Boltzmann transport theory, approximations to the transport equation. Next we introduce several transport methods. We present a section on Caseology, observing transport boundary layers. We briefly broach topics of software development, including verification and validation, and we close with a section on high energy-density experiments that highlight and support radiation transport.
Energy Technology Data Exchange (ETDEWEB)
Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M. [KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven (Belgium); Dekeyser, W. [ITER Organization, route de Vinon-sur-Verdon, 13067 St. Paul lez Durance Cedex (France); Samaey, G. [KU Leuven, Department of Computer Science, Celestijnenlaan 200A, 3001 Leuven (Belgium)
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
International Nuclear Information System (INIS)
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation
Compressible turbulence transport equations for generalized second order closure
Energy Technology Data Exchange (ETDEWEB)
Cloutman, L D
1999-05-01
Progress on the theory of second order closure in turbulence models of various types requires knowledge of the transport equations for various turbulence correlations. This report documents a procedure that provides such equations for a wide variety of turbulence averages for compressible flows of a multicomponent fluid. Generalizing some work by Germano for incompressible flows, we introduce an appropriate extension of his generalized second order correlations and use a generalized mass-weighted averaging procedure to derive transport equations for the correlations. The averaging procedure includes all of the commonly used averages as special cases. The resulting equations provide an internally consistent starting point for future work in developing single-point statistical turbulence transport models for fluid flows. The form invariance of the in-compressible equations also holds for the compressible case, and we discuss some of the closure issues and frequently ignored complications of statistical turbulence models of compressible flows.
UPWIND DISCONTINUOUS GALERKIN METHODS FOR TWO DIMENSIONAL NEUTRON TRANSPORT EQUATIONS
Institute of Scientific and Technical Information of China (English)
袁光伟; 沈智军; 闫伟
2003-01-01
In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied.The stability for both of the semi-discrete and full-discrete method will be proved.
SPECTRAL FINITE ELEMENT METHOD FOR A UNSTEADY TRANSPORT EQUATION
Institute of Scientific and Technical Information of China (English)
MeiLiquan
1999-01-01
In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.
Development of interfacial area transport equation - modeling and experimental benchmark
International Nuclear Information System (INIS)
A dynamic treatment of interfacial area concentration has been studied over the last decade by employing the interfacial area transport equation. When coupled with the two-fluid model, the interfacial area transport equation replaces the flow regime dependent correlations for interfacial area concentration and eliminates potential artificial bifurcation or numerical oscillations stemming from these static correlations. An extensive database has been established to evaluate the model under various two-phase flow conditions. These include adiabatic and heated conditions, vertical and horizontal flow orientations, round, rectangular, annulus and 8×8 rod bundle channel geometries, and normal-gravity and simulated reduced-gravity conditions. This paper reviews the current state-of-the-art in the development of the interfacial area transport equation, available experimental databases and 1D and 3D benchmarking work of the interfacial area transport equation. (author)
Puglisi, Andrea
2015-01-01
This brief offers a concise presentation of granular fluids from the point of view of non-equilibrium statistical physics. The emphasis is on fluctuations, which can be large in granular fluids due to the small system size (the number of grains is many orders of magnitude smaller than in molecular fluids). Firstly, readers will be introduced to the most intriguing experiments on fluidized granular fluids. Then granular fluid theory, which goes through increasing levels of coarse-graining and emerging collective phenomena, is described. Problems and questions are initially posed at the level of kinetic theory, which describes particle densities in full or reduced phase-space. Some answers become clear through hydrodynamics, which describes the evolution of slowly evolving fields. Granular fluctuating hydrodynamics, which builds a bridge to the most recent results in non-equilibrium statistical mechanics, is also introduced. Further and more interesting answers come when the dynamics of a massive intruder are...
STABILITY OF P2 METHODS FOR NEUTRON TRANSPORT EQUATIONS
Institute of Scientific and Technical Information of China (English)
袁光伟; 沈智军; 沈隆钧; 周毓麟
2002-01-01
In this paper the P2 approximation to the one-group planar neutron transport theory is discussed. The stability of the solutions for P2 equations with general boundary conditions, including the Marshak boundary condition, is proved. Moreover,the stability of the up-wind difference scheme for the P2 equation is demonstrated.
Renormalized transport equations for the bistable potential model
Weidlich, Wolfgang; Grabert, Hermann
1980-09-01
Renormalized transport equations for general Fokker-Planck systems are derived and applied to the bistable potential model. The exact equation for the expectation value t can be evaluated in both domains ∈ x ± and ∈ D 0 outside and between the potential minima, leading to drastic differences of the dynamics prevailing in D ± and D 0, respectively.
A Boltzmann-weighted hopping model of charge transport in organic semicrystalline films
Kwiatkowski, Joe J.
2011-01-01
We present a model of charge transport in polycrystalline electronic films, which considers details of the microscopic scale while simultaneously allowing realistically sized films to be simulated. We discuss the approximations and assumptions made by the model, and rationalize its application to thin films of directionally crystallized poly(3-hexylthiophene). In conjunction with experimental data, we use the model to characterize the effects of defects in these films. Our findings support the hypothesis that it is the directional crystallization of these films, rather than their defects, which causes anisotropic mobilities. © 2011 American Institute of Physics.
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Energy Technology Data Exchange (ETDEWEB)
Fidler, Christian
2011-12-16
Polarisation and Nongaussianity are expected to play a central role in future studies of the cosmic microwave background radiation. Polarisation can be split into a divergence-like E-mode and a curl-like B-mode, of which the later can only be induced by primordial gravitational waves (tensor fluctuations of the metric) at leading order. Nongaussianity is not generated at first order and is directly proportional to the primordial Nongaussianity of inflation. Thus B-mode polarisation and Nongaussianity constrain inflation models directly. While E-mode polarisation has already been detected and is being observed with increasing precision, B-mode polarisation and Nongaussianity remains elusive. The absence of B-mode polarisation when the primordial fluctuations are purely scalar holds, however, only in linear perturbation theory. B-mode polarisation is also generated from scalar sources in second order, which may constitute an important background to the search for primordial gravitational waves. While such an effect would naturally be expected to be relevant at tensor-to-scalar ratios of order 10{sup -5}, which is the size of perturbations in the microwave background, only a full second order calculation can tell whether there are no enhancements. For Nongaussianity the situation is analogous: At second order intrinsic Nongaussianities are induced to the spectrum, which may be an important background to the primordial Nongaussianity. After the full second-order Boltzmann equations for the cosmological evolution of the polarised radiation distribution have become available, I focused on the novel sources to B-mode polarisation that appear in the second-order collision term, which have not been calculated before. In my PHD thesis I developed a numerical code, which solves the second order Boltzmann hierarchy and calculates the C{sub l}{sup BB}-spectrum.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
International Nuclear Information System (INIS)
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs
A Generalized Boltzmann Fokker-Planck Method for Coupled Charged Particle Transport
Energy Technology Data Exchange (ETDEWEB)
Prinja, Anil K
2012-01-09
The goal of this project was to develop and investigate the performance of reduced-physics formulations of high energy charged particle (electrons, protons and heavier ions) transport that are computationally more efficient than not only analog Monte Carlo methods but also the established condensed history Monte Carlo technique. Charged particles interact with matter by Coulomb collisions with target nuclei and electrons, by bremsstrahlung radiation loss and by nuclear reactions such as spallation and fission. Of these, inelastic electronic collisions and elastic nuclear collisions are the dominant cause of energy-loss straggling and angular deflection or range straggling of a primary particle. These collisions are characterized by extremely short mean free paths (sub-microns) and highly peaked, near-singular differential cross sections about forward directions and zero energy loss, with the situation for protons and heavier ions more extreme than for electrons. For this reason, analog or truephysics single-event Monte Carlo simulation, while possible in principle, is computationally prohibitive for routine calculation of charged particle interaction phenomena.
Properties of an affine transport equation and its generalized holonomy
Vines, Justin
2014-01-01
We investigate properties of a transport equation that was recently used to study the observer dependence of angular momentum in general relativity. The associated map between the tangent spaces at two points on a curve is affine, and for this reason, the operation was called "affine transport". The map consists of a homogeneous (linear) part given by the parallel transport map along the curve, plus an inhomogeneous part which is related to the development of a curve in a manifold into an affine tangent space (also described as the rolling of a manifold along a tangent space without slipping or twisting). For closed curves, the affine transport equation defines a "generalized holonomy". We use covariant bitensor calculus to compute the generalized holonomy around geodesic polygon loops, specifically for triangles and "parallelogramoids" with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order holonomy of parallel transport ($\\sim$ Riemann $\\times$ area)...
Reciprocal Symmetric Boltzmann Function and Unified Boson-Fermion Statistics
Ahmad, Mushfiq; Talukder, Muhammad O. G.
2007-01-01
The differential equation for Boltzmann's function is replaced by the corresponding discrete finite difference equation. The difference equation is, then, symmetrized so that the equation remains invariant when step d is replaced by -d. The solutions of this equation come in Boson-Fermion pairs. Reciprocal symmetric Boltzmann's function, thus, unifies both Bosonic and Fermionic distributions.
Energy Technology Data Exchange (ETDEWEB)
Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Dept. and CEEM, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [Physics Dept., Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)
2014-11-15
To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study.
International Nuclear Information System (INIS)
To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study
International Nuclear Information System (INIS)
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems reach the onset of Bose–Einstein condensation in a finite time. The approach to condensation is characterized by a scaling behavior that we briefly analyze
Energy Technology Data Exchange (ETDEWEB)
Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Department and Center for Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)
2013-12-20
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems reach the onset of Bose–Einstein condensation in a finite time. The approach to condensation is characterized by a scaling behavior that we briefly analyze.
Thermodynamic framework for a generalized heat transport equation
Directory of Open Access Journals (Sweden)
Guo Yangyu
2016-06-01
Full Text Available In this paper, a generalized heat transport equation including relaxational, nonlocal and nonlinear effects is provided, which contains diverse previous phenomenological models as particular cases. The aim of the present work is to establish an extended irreversible thermodynamic framework, with generalized expressions of entropy and entropy flux. Nonlinear thermodynamic force-flux relation is proposed as an extension of the usual linear one, giving rise to the nonlinear terms in the heat transport equation and ensuring compatibility with the second law. Several previous results are recovered in the linear case, and some additional results related to nonlinear terms are also obtained.
Renormalization of transport equations in Fokker-Planck models
Grabert, Hermann; Weidlich, Wolfgang
1980-06-01
This paper is concerned with the derivation of nonlinear fluctuation-renormalized transport equations of a fluctuating thermodynamic system, on the assumption that the macroscopic variables defining a state undergo a Fokker-Planck process. It is shown that the renormalization effect may consist of two parts: a renormalization of the thermodynamic forces and a renormalization of the transport coefficients. Closed analytical expressions for the renormalized quantities in terms of the bare quantities appearing in the Fokker-Planck equation are derived. A scheme for the approximate evaluation of these expressions is given.
International Nuclear Information System (INIS)
Neutron transport in nuclear reactors is quite well modelled by the linear Boltzmann transport equation. Its solution is relatively easy, but unfortunately too expensive to achieve whole core computations. Thus, we have to simplify it, for example by homogenizing some physical characteristics. However, the solution may then be inaccurate. Moreover, in strongly homogeneous areas, the error may be too big. Then we would like to deal with such an inconvenient by solving the equation accurately on this area, but more coarsely away from it, so that the computation is not too expensive. This problem is the subject of a thesis. We present here some results obtained for slab geometry. The couplings between the fine and coarse discretization regions could be conceived in a number of approaches. Here, we only deal with the coupling at crossing the interface between two sub-domains. In the first section, we present the coupling of discrete ordinate methods for solving the homogeneous, isotropic and mono-kinetic equation. Coupling operators are defined and shown to be optimal. The second and the third sections are devoted to an extension of the previous results when the equation is non-homogeneous, anisotropic and multigroup (under some restrictive assumptions). Some numerical results are given in the case of isotropic and mono-kinetic equations. (author)
A rain splash transport equation assimilating field and laboratory measurements
Dunne, T.; Malmon, D.V.; Mudd, S.M.
2010-01-01
Process-based models of hillslope evolution require transport equations relating sediment flux to its major controls. An equation for rain splash transport in the absence of overland flow was constructed by modifying an approach developed by Reeve (1982) and parameterizing it with measurements from single-drop laboratory experiments and simulated rainfall on a grassland in East Africa. The equation relates rain splash to hillslope gradient, the median raindrop diameter of a storm, and ground cover density; the effect of soil texture on detachability can be incorporated from other published results. The spatial and temporal applicability of such an equation for rain splash transport in the absence of overland flow on uncultivated hillslopes can be estimated from hydrological calculations. The predicted transport is lower than landscape-averaged geologic erosion rates from Kenya but is large enough to modify short, slowly eroding natural hillslopes as well as microtopographic interrill surfaces between which overland flow transports the mobilized sediment. Copyright 2010 by the American Geophysical Union. Copyright 2010 by the American Geophysical Union.
Properties of an affine transport equation and its holonomy
Vines, Justin; Nichols, David A.
2016-10-01
An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transport equation defines a "generalized holonomy" that takes the form of an affine map on the tangent space. We explore the local properties of this generalized holonomy by using covariant bitensor methods to compute the generalized holonomy around geodesic polygon loops. We focus on triangles and "parallelogramoids" with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order linear holonomy (˜ Riemann × area), and we derive the leading-order inhomogeneous part of the generalized holonomy (˜ Riemann × area^{3/2}). Our bitensor methods let us naturally compute higher-order corrections to these leading results. These corrections reveal the form of the finite-size effects that enter into the holonomy for larger loops; they could also provide quantitative errors on the leading-order results for finite loops.
Energy Technology Data Exchange (ETDEWEB)
Kawakami, H.; Urabe, J.; Yukimura, K. (Doshisha Univ., Kyoto (Japan))
1991-03-20
In a discharge excitation rare gas halide excima laser, uniform generation and stable maintenance of the excited discharge determines the laser characteristics. In this report, an approximate solution was obtained on the Boltzmann equation (frequently used for the theoretical analysis of this laser) to examine the nature of the solution. By optimizing the conversion of the variables, calculation of an electron swarm parameter in the hitherto uncertain range of the low conversion electric field was made possible, giving a generation mechanism of the uncertainty of the excited dischareg. The results are summarized as below. (1) The Boltzmann equation gives a linear solution for a logarithmic value of an electron energy in the range of low conversion electric field. (2) Time-wise responce ability between the measured voltage, current characteristics of the excitation discharge was clarified and the attachment and ionization coefficients calculated by Boltzmann equation. (3) Dependency of the attachment coefficient on the partial pressure of fluorine and kripton was examined, and the attachment coefficient was found to increase with the increase of the partial pressure for the both cases. 20 refs., 9 figs., 2 tabs.
Deng, Yun-Kun; Xiao, Deng-Ming
2013-03-01
The electron swarm parameters including the density-normalized effective ionization coefficients (α-η)/N and the electron drift velocities Ve are calculated for a gas mixture of CF3I with N2 and CO2 by solving the Boltzmann equation in the condition of a steady-state Townsend (SST) experiment. The overall density-reduced electric field strength is from 100 Td to 1000 Td (1 Td = 10-17 V·cm2), while the CF3I content k in the gas mixture can be varied over the range from 0% to 100%. From the variation of (α-η)/N with the CF3I mixture ratio k, the limiting field strength (E/N)lim for each CF3I concentration is derived. It is found that for the mixtures with 70% CF3I, the values of (E/N)lim are essentially the same as that for pure SF6. Additionally, the global warming potential (GWP) and the liquefaction temperature of the gas mixtures are also taken into account to evaluate the possibility of application in the gas insulation of power equipment.
Deng, Yunkun; Xiao, Dengming
2014-09-01
The present study is devoted to the calculation of electron swarm parameters, including the reduced effective ionization coefficient, electron mean energy, and electron drift velocity, for the gas mixtures of CF3I with Ar, Xe, He, N2, and CO2. These data are computed by employing the Boltzmann equation method with two-term approximation in the condition of steady-state Townsend (SST) discharge. For the purpose of evaluating the insulation strength of CF3I gas mixtures, values of the limiting field strength (E/N)lim for which the ionization exactly balances the electron attachment are determined from the variation curves of (α - η)/N. The results indicate that mixtures of CF3I-N2 present the greatest insulation strength among all the combinations for CF3I content varied from 20 to 90%. Furthermore, the gas mixture with 70% CF3I can achieve a very similar dielectric strength to that of SF6. The concerned liquefaction issues are also taken into account to fully assess the possibility of applying CF3I gas mixtures in power equipment as an insulation medium.
Institute of Scientific and Technical Information of China (English)
Deng Yun-Kun; Xiao Deng-Ming
2013-01-01
The electron swarm parameters including the density-normalized effective ionization coefficients (α-η)/N and the electron drift velocities Ve are calculated for a gas mixture of CF3I with N2 and CO2 by solving the Boltzmann equation in the condition of a steady-state Townsend (SST) experiment.The overall density-reduced electric field strength is from 100 Td to 1000 Td (1 Td =10-17 V·cm2),while the CF3I content k in the gas mixture can be varied over the range from 0％ to 100％.From the variation of (α-η)/N with the CF3I mixture ratio k,the limiting field strength (E/N)lim for each CF3I concentration is derived.It is found that for the mixtures with 70％ CF3I,the values of (E/N)lim are essentially the same as that for pure SF6.Additionally,the global warming potential (GWP) and the liquefaction temperature of the gas mixtures are also taken into account to evaluate the possibility of application in the gas insulation of power equipment.
Energy Technology Data Exchange (ETDEWEB)
Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Transport equations for a general class of evolution equations with random perturbations
Guo, Maozheng; Wang, Xiao-Ping
1999-10-01
We derive transport equations from a general class of equations of form iut=H(X,D)u+V(X,D)u where H(X,D) and V(X,D) are pseudodifferential operators (Weyl operator) with symbols H(x,k) and V(x,k), where H(x,k) being polynomial in k and smooth in x,V(x,k) is a mean zero random function and is stationary in space variable. We also consider system of equations in the above form. Such equations cover many of the equations that arise in wave propagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327-370 (1996)]. Our results generalize those by Ryzhik, Papanicolau, and Keller.
Transport modelling in coastal waters using stochastic differential equations
Charles, W.M.
2007-01-01
In this thesis, the particle model that takes into account the short term correlation behaviour of pollutants dispersion has been developed. An efficient particle model for sediment transport has been developed. We have modified the existing particle model by adding extra equations for the suspensio
MSSM electroweak baryogenesis and flavour mixing in transport equations
Energy Technology Data Exchange (ETDEWEB)
Konstandin, Thomas [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: t.konstandin@thphys.uni-heidelberg.de; Prokopec, Tomislav [Institute for Theoretical Physics (ITF) and Spinoza Institute, Utrecht University, Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht (Netherlands)]. E-mail: t.prokopec@phys.uu.nl; Schmidt, Michael G. [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: m.g.schmidt@thphys.uni-heidelberg.de; Seco, Marcos [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: m.seco@thphys.uni-heidelberg.de
2006-03-20
We make use of the formalism of [T. Konstandin, et al., hep-ph/0410135], and calculate the chargino-mediated baryogenesis in the Minimal Supersymmetric Standard Model. The formalism makes use of a gradient expansion of the Kadanoff-Baym equations for mixing fermions. For illustrative purposes, we first discuss the semiclassical transport equations for mixing bosons in a space-time-dependent Higgs background. To calculate the baryon asymmetry, we solve a standard set of diffusion equations, according to which the chargino asymmetry is transported to the top sector, where it biases sphaleron transitions. At the end we make a qualitative and quantitative comparison of our results with the existing work. We find that the production of the baryon asymmetry of the universe by CP-violating currents in the chargino sector is strongly constrained by measurements of electric dipole moments.
Transport coefficients for inelastic Maxwell mixtures
Garzo, Vicente; Astillero, Antonio
2004-01-01
The Boltzmann equation for inelastic Maxwell models is used to determine the Navier-Stokes transport coefficients of a granular binary mixture in $d$ dimensions. The Chapman-Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transpor...
Dyatko, Nikolay; Donkó, Zoltán
2015-08-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This ‘bistability effect’—in which electron-electron (Coulomb) collisions play an essential role—is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms (‘two-term approximation’), (ii) neglecting the Coulomb part of the collision integral for the anisotropic part of the distribution function, (iii) treating Coulomb collisions as binary events, and (iv) truncating the range of the electron-electron interaction beyond a characteristic distance. The particle-based simulation method avoids these approximations: the many-body interactions within the electron gas with a true (un-truncated) Coulomb potential are described by a molecular dynamics algorithm, while the collisions between electrons and the background gas atoms are treated with Monte Carlo simulation. We find a good general agreement between the results of the two techniques, which confirms, to a certain extent, the approximations used in the solution of the Boltzmann equation. The differences observed between the results are believed to originate from these approximations and from the presence of statistical noise in the particle simulations.
International Nuclear Information System (INIS)
We present a scheme for numerical simulations of collisionless self-gravitating systems which directly integrates the Vlasov-Poisson equations in six-dimensional phase space. Using the results from a suite of large-scale numerical simulations, we demonstrate that the present scheme can simulate collisionless self-gravitating systems properly. The integration scheme is based on the positive flux conservation method recently developed in plasma physics. We test the accuracy of our code by performing several test calculations, including the stability of King spheres, the gravitational instability, and the Landau damping. We show that the mass and the energy are accurately conserved for all the test cases we study. The results are in good agreement with linear theory predictions and/or analytic solutions. The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are run on 646 grids. The computation speed scales well with the number of processors, and thus our code performs efficiently on massively parallel supercomputers.
An anisotropic scattering treatment for the even parity transport equation
International Nuclear Information System (INIS)
This work introduces an extension of the even-odd parity flux formulation to the treatment of anisotropic scattering with the finite element formulation. To keep a similar form to the even parity transport equation in the isotropic case, we define a 'direction-dependent cross section'. We have only two unkown functions (the even parity flux, and the direction dependent cross section), that are calculated via an iterative process. We consider the multigroup equation for the eigenvalue problem and we present some numerical tests to prove the effectiveness of this method. (orig.)
The Einstein-Boltzmann system and positivity
Lee, Ho
2012-01-01
The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.
Moment transport equations and their application to the perturbed universe
Sierra, Carlos A
2015-01-01
There are many inflationary models that allow the formation of the large-scale structure of the observable universe. The non-gaussianity parameter $f_{NL}$ is a useful tool to discriminate among these cosmological models when comparing the theoretical predictions with the satellite survey results like those from WMAP and Planck. The goal of this proceeding contribution is to review the moment transport equations methodology and the subsequent calculation of the $f_{NL}$ parameter.
Fundamental solution of an integro-differential transport equation
International Nuclear Information System (INIS)
The problems connected with the construction of generalized functions of the fundamental solution of a stationary single-velocity integro-differential neutron transport equation on the basis of the integral transformation technique and its applications are analyzed. A number of concrete expressions for the fundamental solutions in the tree-dimensional (spherical coordinates) and two-dimensional (cylindrical coordinates) geometries are derived. The most of the formula are apparently published for the first time
Extension to the integral transport equation of an iterative method
Energy Technology Data Exchange (ETDEWEB)
Jehouani, A. [EPRA, Department of Physics, Faculty of Sciences, Semlalia, PO Box 2390, 40000 Marrakech (Morocco)]. E-mail: jehouani@ucam.ac.ma; Elmorabiti, A. [Centre d' Etudes Nucleaires de Maamoura, CNESTEN (Morocco); Ghassoun, J. [EPRA, Department of Physics, Faculty of Sciences, Semlalia, PO Box 2390, 40000 Marrakech (Morocco)
2006-09-15
In this paper we describe an extension to the neutron integral transport equation of an iterative method. Indeed an iterative scheme is used for both energy and space including external iteration for the multiplication factor and internal iteration for flux calculations. The Monte Carlo method is used to evaluate the spatial transfer integrals. The results were compared with those obtained by using the APOLLO2 code for a cylindrical cell.
Extension to the integral transport equation of an iterative method
International Nuclear Information System (INIS)
In this paper we describe an extension to the neutron integral transport equation of an iterative method. Indeed an iterative scheme is used for both energy and space including external iteration for the multiplication factor and internal iteration for flux calculations. The Monte Carlo method is used to evaluate the spatial transfer integrals. The results were compared with those obtained by using the APOLLO2 code for a cylindrical cell
Comparison of neutronic transport equation resolution nodal methods
International Nuclear Information System (INIS)
In this work, some transport equation resolution nodal methods are comparatively studied: the constant-constant (CC), linear-nodal (LN) and the constant-quadratic (CQ). A nodal scheme equivalent to finite differences has been used for its programming, permitting its inclusion in existing codes. Some bidimensional problems have been solved, showing that linear-nodal (LN) are, in general, obtained with accuracy in CPU shorter times. (Author)
Deterministic methods to solve the integral transport equation in neutronic
International Nuclear Information System (INIS)
We present a synthesis of the methods used to solve the integral transport equation in neutronic. This formulation is above all used to compute solutions in 2D in heterogeneous assemblies. Three kinds of methods are described: - the collision probability method; - the interface current method; - the current coupling collision probability method. These methods don't seem to be the most effective in 3D. (author). 9 figs
Deterministic methods to solve the differential transport equation in neutronic
International Nuclear Information System (INIS)
We present a synthesis of the methods used to solve the integro-differential form of the transport equation. This form is used above all to achieve whole core calculations in 2 and 3D. First, we discretize the equation in energy and it leads us to an one group energy equation. For each of these groups, the scope of the calculation is so big that we have to calculate our solution on homogenized cells. On this homogenized medium, we describe different angular and spatial discretizations with acceleration methods. Finally we select some promising schemes to test: - SN Linear Nodal method with a Diffusion Synthetic Acceleration method; - Variational Nodal Method. These methods could be compared with more classical ones. To say, finite element or finite difference methods. (author). 65 refs., 3 annexes
Lattice Boltzmann method with the cell-population equilibrium
Institute of Scientific and Technical Information of China (English)
Zhou Xiao-Yang; Cheng Bing; Shi Bao-Chang
2008-01-01
The central problem of the lattice Boltzmann method (LBM) is to construct a discrete equilibrium.In this paper,a multi-speed 1D cell-model of Boltzmann equation is proposed,in which the cell-population equilibrium,a direct nonnegative approximation to the continuous Maxwellian distribution,plays an important part.By applying the explicit one-order Chapman-Enskog distribution,the model reduces the transportation and collision,two basic evolution steps in LBM,to the transportation of the non-equilibrium distribution.Furthermore,1D dam-break problem is performed and the numerical results agree well with the analytic solutions.
Relativistic Entropy and Related Boltzmann Kinetics
Kaniadakis, G
2009-01-01
It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fix univocally the entropy of the system, which turn out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitely remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle, suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativ...
Heavy Flavor Suppression: Boltzmann vs Langevin
Das, Santosh K; Plumari, Salvatore; Greco, Vincenzo
2013-01-01
The propagation of heavy flavor through the quark gluon plasma has been treated commonly within the framework of Langevin dynamics, i.e. assuming the heavy flavor momentum transfer is much smaller than the light one. On the other hand a similar suppression factor $R_{AA}$ has been observed experimentally for light and heavy flavors. We present a thorough study of the approximations involved by Langevin equation by mean of a direct comparison with the full collisional integral within the framework of Boltzmann transport equation. We have compared the results obtained in both approaches which can differ substantially for charm quark leading to quite different values extracted for the heavy quark diffusion coefficient. In the case of bottom quark the approximation appears to be quite reasonable.
Stochastic equations in the invariant imbedding formulation of particle transport
International Nuclear Information System (INIS)
Invariant imbedding theory is an alternative formulation of particle transport theory. Although stochastic foundations of invariant imbedding have been known from the beginnings, the method itself has so far exclusively been used for calculating first moments, i.e. expectations. The present paper attempts to set up a probability balance equation in the invariant imbedding approach from which equations for the first and second order densities are derived. It is shown that only the equations for the first order densities are non-linear, while subsequent order densities obey linear equations. This is expected to considerably simplify solution to those problems which involve second order density calculations where invariant imbedding techniques may be profitably used. Examples of such quantities are the variance or correlations between particles detected at two different energies or angles or the higher moments of the emitted multiplicity distribution such as the variance from a target bombarded by incident particles. One possible area of application of our equations is non-destructive estimation of fissile material by the active neutron assay technique. Another area is the study of particle cascade development in sputtering and positron backscattering from surfaces. The approach is illustrated by a simple forward-backward scattering model for these two problems.
Guo, Pengfei; Guan, Yong; Liu, Gang; Liang, Zhiting; Liu, Jianhong; Zhang, Xiaobo; Xiong, Ying; Tian, Yangchao
2016-09-01
This work reports an investigation of the impact of microstructure on the performance of solid oxide fuel cells (SOFC) composed of nickel yttria-stabilized zirconia (Ni YSZ). X-ray nano computed tomography (nano-CT) was used to obtain three-dimensional (3D) models of Ni-YSZ composite anode samples subjected to different thermal cycles. Key parameters, such as triple phase boundary (TPB) density, were calculated using 3D reconstructions. The electrochemical reaction occurring at active-TPB was modeled by the Lattice Boltzmann Method for simulation of multi-component mass transfer in porous anodes. The effect of different electrode geometries on the mass transfer and the electrochemical reaction in anodes was studied by TPB distributions measured by nano CT for samples subjected to different thermal cycles. The concentration polarization and the activation polarization were estimated respectively. The results demonstrate that a combined approach involving nano-CT experiments in conjunction with simulations of gas transport and electrochemical reactions using the Lattice Boltzmann method can be used to better understand the relationship between electrode microstructure and performance of nickel yttria-stabilized zirconia anodes.
Application of the Broyden method to stiff transport equations
Carlsson, Johan; Cary, John R.; Cohen, Ron
2002-11-01
Plasma turbulence generates fluxes (of particles, energy, etc.) that are said to be stiff, that is a small change in temperature, density, or some other quantity, can lead to a large change in flux. The dependence of the diffusivities on the temperature and density profiles, and their gradients, also introduces nonlinearity. Irrespective of whether the fluxes are given by transport models, such as IFS/PPPL, GLF23, or MMM95, or are directly calculated, the resulting system of transport equations is thus numerically challenging to solve. Efficient transport solvers must also take into account that the evaluation of the diffusivities (or their gradients: the fluxes) is numerically costly. We have developed a new iterative transport solver that combines the stability of a relaxation scheme with the fast convergence of a Newton solver. The new solver uses a gradually decreasing relaxation parameter for the first few iterations and once it is inside the radius of convergence it switches over to a quasi-Newton method where a Broyden-like scheme is used to approximate the Jacobian. By taking advantage of the structure of the matrix (tri-diagonal if a second-order spatial finite differencing is used) the Broyden algorithm[1] gives a good approximation of the Jacobian after only a few iterations. We have implemented the new transport solver in the form of a C++ library called the Transport Analysis Tool. To make the library easy to access from codes written in other languages, a C interface is also provided. We will present the new transport solver in detail, as well as benchmark results and examples of how to use the Transport Analysis Tool library. [1] C. G. Broyden, in Mathematics of Computation, vol. 19, 1965, pp. 577--593.
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations
Brenier, Yann
2009-10-01
We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61-97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge-Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375-417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141-1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131-151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20-31, 1991, Arch. Ration. Mech. Anal. 185:341-363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450-1461, 1998; Loeper in SIAM J. Math. Anal. 38:795-823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov-Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29-59, 2001) and of the Keller-Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225-234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819-824, 1992; Chalub et al. in Mon. Math. 142:123-141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier-Stokes-Boussinesq equations
Energy Technology Data Exchange (ETDEWEB)
Uchida, S.; Sugawara, H.; Ventzek, P.; Sakai, Y. [Hokkaido University, Sapporo (Japan)
1998-06-01
Xe/Ne plasmas are important for plasma display panels and VUV light sources. However, reactions between electrons and excited particles in the mixtures are so complicated that influence of the reactions on the plasma properties is not understood well. In this work, taking account of reactions through which electrons are produced, such as cumulative and Penning ionization, and of transition between excited levels, the electron and excited particle properties in Xe/Ne plasmas are calculated using the Boltzmann equation. The ionization coefficient and electron drift velocity agreed with experimental data. The influence of laser absorption in Xe/Ne plasmas on the plasma properties is also discussed. 25 refs., 15 figs.
Monte Carlo solution of a semi-discrete transport equation
International Nuclear Information System (INIS)
The authors present the S∞ method, a hybrid neutron transport method in which Monte Carlo particles traverse discrete space. The goal of any deterministic/stochastic hybrid method is to couple selected characters from each of the methods in hopes of producing a better method. The S∞ method has the features of the lumped, linear-discontinuous (LLD) spatial discretization, yet it has no ray-effects because of the continuous angular variable. They derive the S∞ method for the solid-state, mono-energetic transport equation in one-dimensional slab geometry with isotropic scattering and an isotropic internal source. They demonstrate the viability of the S∞ method by comparing their results favorably to analytic and deterministic results
Transport methods: general. 8. Formulation of Transport Equation in a Split Form
International Nuclear Information System (INIS)
The singular eigenfunction expansion method has enabled the application of functional analysis methods in transport theory. However, when applying it, the users were discouraged, since in most problems, including slab problems, an extra problem has occurred. It appears necessary to solve the Fredholm integral equation in order to determine the expansion coefficients. There are several reasons for this difficulty. One reason might be the use of the full-range expansion techniques even in the regions where the function is singular. Such an example is the free boundary condition that requires the distribution to be equal to zero. Moreover, at μ = 0, the transport equation becomes an integral one. Both reasons motivated us to redefine the transport equation in a more natural way. Similar to scattering theory, here we define the flux distribution as a direct sum of forward- and backward-directed neutrons, e.g., μ ≥ 0 and μ < 0, respectively. As a result, the plane geometry transport equation is being split into coupled-pair equations. Further, using an appropriate transformation, this pair of equations reduces to a self-adjoint one having the same form as the known full-range single flux. It is interesting that all the methods of full-range theory are applicable here provided the flux as well as the transformed transport operator are two-dimensional matrices. Applying this to the slab problem, we find explicit expressions for reflected and transmitted particles caused by an arbitrary plane source. That is the news in this paper. Because of space constraints, only fundamentals of this approach will be presented here. We assume that the reader is familiar with this field; therefore, the applications are noted only at the end. (author)
Morales-Casique, E.; Lezama-Campos, J. L.; Guadagnini, A.; Neuman, S. P.
2013-05-01
Modeling tracer transport in geologic porous media suffers from the corrupt characterization of the spatial distribution of hydrogeologic properties of the system and the incomplete knowledge of processes governing transport at multiple scales. Representations of transport dynamics based on a Fickian model of the kind considered in the advection-dispersion equation (ADE) fail to capture (a) the temporal variation associated with the rate of spreading of a tracer, and (b) the distribution of early and late arrival times which are often observed in field and/or laboratory scenarios and are considered as the signature of anomalous transport. Elsewhere we have presented exact stochastic moment equations to model tracer transport in randomly heterogeneous aquifers. We have also developed a closure scheme which enables one to provide numerical solutions of such moment equations at different orders of approximations. The resulting (ensemble) average and variance of concentration fields were found to display a good agreement against Monte Carlo - based simulation results for mildly heterogeneous (or well-conditioned strongly heterogeneous) media. Here we explore the ability of the moment equations approach to describe the distribution of early arrival times and late time tailing effects which can be observed in Monte-Carlo based breakthrough curves (BTCs) of the (ensemble) mean concentration. We show that BTCs of mean resident concentration calculated at a fixed space location through higher-order approximations of moment equations display long tailing features of the kind which is typically associated with anomalous transport behavior and are not represented by an ADE model with constant dispersive parameter, such as the zero-order approximation.
Propagator solutions of partial differential equations applied to plasma transport
International Nuclear Information System (INIS)
A variety of partial differential equations may be solved efficiently and easily, using a single algorithm which is based on the propagator (or Green's function) for equations of the Fokker-Planck type. The numerical method employed is always at least as fast as finite differencing, but is especially efficient in cases where flow dominates over diffusion, and is consequently potentially widely applicable in fluid dynamics. Two transport problems in a small toroidal magnetic confinement device (IMS) have been considered. First, the ion distribution function and the potential in the sheath were calculated in the presence of charge exchange collisions for a wide range of neutral densities. Second, radial particle transport was modeled allowing for strong convection in the bulk plasma, and the density n(r,θ) was obtained for various plausible forms of the flow velocity, consistent with the observed average radial velocity. The best agreement with experiment is obtained with a vertical flow, which is also suggested by the measured potential profiles. 17 refs., 10 figs., 1 tab
Boltzmann's Concept of Reality
Ribeiro, Marcelo B.; Videira, Antonio A. P.
2007-01-01
In this article we describe and analyze the concept of reality developed by the Austrian theoretical physicist Ludwig Boltzmann. It is our thesis that Boltzmann was fully aware that reality could, and actually was, described by different points of view. In spite of this, Boltzmann did not renounce the idea that reality is real. We also discuss his main motivations to be strongly involved with philosophy of science, as well as further developments made by Boltzmann himself of his main philosop...
International Nuclear Information System (INIS)
This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)
A unified transport equation for both cosmic rays and thermal particles
Williams, L. L.; Schwadron, N.; Jokipii, J. R.; Gombosi, T. I.
1993-01-01
We present a unified transport equation that is valid for particles of all energies if the particle mean free paths are much smaller than macroscopic fluid length scales. If restricted to particles with random speeds much greater than fluid flow speeds, this equation reduces to the previously discussed extended cosmic-ray transport equation. It is significant that this allows one to describe the acceleration of particles from thermal energies to cosmic-ray energies using one transport equation. This is in contrast to previous transport equations (the Parker equation and the extended cosmic-ray transport equation), which were restricted to fast particles. The close connection to the extended cosmic-ray transport equation is demonstrated.
Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture
Davey, K.; Darvizeh, R.
2016-09-01
Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine-Hugoniot conditions for fracture are established along with extended forms of J -integrals.
Energy Technology Data Exchange (ETDEWEB)
Li, M
1998-08-01
In this thesis, two methods for solving the multigroup Boltzmann equation have been studied: the interface-current method and the Monte Carlo method. A new version of interface-current (IC) method has been develop in the TDT code at SERMA, where the currents of interface are represented by piecewise constant functions in the solid angle space. The convergence of this method to the collision probability (CP) method has been tested. Since the tracking technique is used for both the IC and CP methods, it is necessary to normalize he collision probabilities obtained by this technique. Several methods for this object have been studied and implemented in our code, we have compared their performances and chosen the best one as the standard choice. The transfer matrix treatment has been a long-standing difficulty for the multigroup Monte Carlo method: when the cross-sections are converted into multigroup form, important negative parts will appear in the angular transfer laws represented by low-order Legendre polynomials. Several methods based on the preservation of the first moments, such as the discrete angles methods and the equally-probable step function method, have been studied and implemented in the TRIMARAN-II code. Since none of these codes has been satisfactory, a new method, the non equally-probably step function method, has been proposed and realized in our code. The comparisons for these methods have been done in several aspects: the preservation of the moments required, the calculation of a criticality problem and the calculation of a neutron-transfer in water problem. The results have showed that the new method is the best one in all these comparisons, and we have proposed that it should be a standard choice for the multigroup transfer matrix. (author) 76 refs.
On standard forms for transport equations and fluxes: Part 2
International Nuclear Information System (INIS)
Quasilinear expressions for anomalous particle and energy fluxes arising from electrostatic plasma turbulence in a tokamak are reviewed yet again. Further clarifications are made, and the position taken in a previous report is modified. There, the total energy flux, Qj, and the conductive heat flux, qj, were correctly defined, and the anomalous Qj was correctly calculated. It was shown that the anomalous energy transport can be correctly described by ∇·Q*j, where Q*j = 3/5 Qj, with all remaining source terms such as left-angle pj∇·Vj} cancelling. Here, a revised discussion is given of the identification of the anomalous conductive flux, qj, in which the distinction between Qj and Q*j is reconsidered. It is shown that there is more than one consistent way to define qj. Transport calculations involving only theoretical electrostatic turbulent fluxes are unaffected by these distinctions since Qj or Q*j, rather than qj, is the quantity naturally calculated in the theory. However, an ambiguity remains in experimental transport analysis if the measured particle flux Γj = njVj is to be used in the energy equation. This is because we cannot be sure how properly to treat the source terms pj ∇·Vj or { pj ∇·Vj}. 17 refs
Energy Technology Data Exchange (ETDEWEB)
Verdu, G. [Departamento de Ingenieria Quimica Y Nuclear, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain); Capilla, M.; Talavera, C. F.; Ginestar, D. [Dept. of Nuclear Engineering, Departamento de Matematica Aplicada, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain)
2012-07-01
PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)
Simplified P$_N$ Equations for Nonclassical Transport with Isotropic Scattering
Vasques, R
2016-01-01
A nonclassical diffusion approximation has been previously derived for the the one-speed nonclassical transport equation with isotropic scattering. In this paper we use an asymptotic analysis to derive more accurate diffusion approximations to the nonclassical transport equation. If the free-path distribution is given by an exponential (classical transport), these approximations reduce to the simplified P$_N$ (SP$_N$) equations; therefore, they are labeled nonclassical SP$_N$ equations.
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion. PMID:26565365
Riccati equation for simulation of leads in quantum transport
Bravi, M.; Farchioni, R.; Grosso, G.; Pastori Parravicini, G.
2014-10-01
We present a theoretical procedure with numerical demonstration of a workable and efficient method to evaluate the surface Green's function of semi-infinite leads connected to a device. Such a problem always occurs in quantum transport calculations but also in the study of surfaces and heterojunctions. We show here that these semi-infinite leads can be properly described by real-energy Green's functions obtained analytically by a smart solution of the Riccati matrix equation. The performance of our method is demonstrated in the case of a multichain two-dimensional electron-gas system, composed of a central ribbon connected to two semi-infinite leads, pierced by two opposite magnetic fields.
A discrete formulation of the Wigner transport equation
Kim, Kyoung-Youm
2007-12-01
A discrete formulation of the Wigner distribution function (WDF) and the Wigner transport equation (WTE) is proposed, where the "discreteness" of the WDF and WTE is not just a practical, mathematical feature of discretization for the possible computations, but reveals a fundamental physics regarding the maximum correlation length of potentials (an essential quantum-mechanical feature of the WTE): it is set by the positional uncertainty due to the discrete values of momentum in evaluating the discrete WDF. Our formulation also shows that the weighting function to the potential-correlation term can be derived naturally from a mathematical necessity related to the antiperiodicity of the discrete density operator. In addition, we propose a mutually independent discretization scheme for the diagonal and cross-diagonal coordinates of the density operator, which results in a numerically effective discrete WTE in that it requires much less computational resources without significant loss in accuracy.
Transport equations for linear surface waves with random underlying flows
Bal, Guillaume; Chou, Tom
1999-11-01
We define the Wigner distribution and use it to develop equations for linear surface capillary-gravity wave propagation in the transport regime. The energy density a(r, k) contained in waves propagating with wavevector k at field point r is given by dota(r,k) + nabla_k[U_⊥(r,z=0) \\cdotk + Ω(k)]\\cdotnabla_ra [13pt] \\: hspace1in - (nabla_r\\cdotU_⊥)a - nabla_r(k\\cdotU_⊥)\\cdotnabla_ka = Σ(δU^2) where U_⊥(r, z=0) is a slowly varying surface current, and Ω(k) = √(k^3+k)tanh kh is the free capillary-gravity dispersion relation. Note that nabla_r\\cdotU_⊥(r,z=0) neq 0, and that the surface currents exchange energy density with the propagating waves. When an additional weak random current √\\varepsilon δU(r/\\varepsilon) varying on the scale of k-1 is included, we find an additional scattering term Σ(δU^2) as a function of correlations in δU. Our results can be applied to the study of surface wave energy transport over a turbulent ocean.
Multicomponent transport in weakly ionized mixtures
Energy Technology Data Exchange (ETDEWEB)
Giovangigli, V [CMAP, CNRS, Ecole Polytechnique, 91128 Palaiseau (France); Graille, B [Laboratoire de Mathematiques, Universite Paris-Sud, 91405, Orsay (France); Magin, T [Center for Turbulence Research, Stanford University, CA 94305 (United States); Massot, M, E-mail: vincent.giovangigli@polytechnique.f [Laboratoire EM2C, Ecole Centrale de Paris, 92295 Chatenay-Malabry (France)
2010-06-15
We discuss transport coefficients in weakly ionized mixtures. We investigate the situations of weak and strong magnetic fields as well as electron temperature nonequilibrium. We present in each regime the Boltzmann equations, examples of transport fluxes, the structure of transport linear systems and discuss their solution by efficient iterative techniques. Numerical simulations are presented for partially ionized high-temperature air.
A kinetic model for chemical reactions without barriers : transport coefficients and eigenmodes
Alves, Giselle M.; Marques Júnior, Wilson; Soares, A. J.; Kremer, Gilberto M.
2011-01-01
The kinetic model of the Boltzmann equation proposed in the work of Kremer and Soares 2009 for a binary mixture undergoing chemical reactions of symmetric type which occur without activation energy is revisited here, with the aim of investigating in detail the transport properties of the reactive mixture and the influence of the reaction process on the transport coefficients. Accordingly, the non-equilibrium solution of the Boltzmann equation is determined through an expansion in Sonine polyn...
Investigation of Resistivity of Saturated Porous Media with Lattice Boltzmann Method
Institute of Scientific and Technical Information of China (English)
YUE Wen-Zheng; TAO Guo; ZHU Ke-Qin
2004-01-01
The lattice Boltzmann method is employed to study the electrical transport properties of saturated porous media.Electrical current flow through the porous media is simulated and the relationship between resistivity index and water saturation is derived. It is found that this kind of relation is not a straight line as described by the Archie equation with the parameter n being a constant in a log-log scale. A new equation is thus developed to formulate this relation with n being a function of porosity and water saturation. The comparisons between the results by lattice Boltzmann and by the laboratory experiments on rock samples demonstrate that this numerical method can provide an alternative way for the expensive laboratory experiments to investigate the electrical transport properties of saturated porous media and can be used to explore micro mechanisms more conveniently.
Influence of asperities on fluid and thermal flow in a fracture: a coupled Lattice Boltzmann study
Neuville, Amélie; Toussaint, Renaud
2013-01-01
The characteristics of the hydro-thermal flow which occurs when a cold fluid is injected into a hot fractured bedrock depend on the morphology of the fracture. We consider a sharp triangular asperity, invariant in one direction, perturbing an otherwise flat fracture. We investigate its influence on the macroscopic hydraulic transmissivity and heat transfer efficiency, at fixed low Reynolds number. In this study, numerical simulations are done with a coupled lattice Boltzmann method that solves both the complete Navier-Stokes and advection-diffusion equations in three dimensions. The results are compared with those obtained under lubrication approximations which rely on many hypotheses and neglect the three-dimensional (3D) effects. The lubrication results are obtained by analytically solving the Stokes equation and a two-dimensional (integrated over the thickness) advection-diffusion equation. We use a lattice Boltzmann method with a double distribution (for mass and energy transport) on hypercubic and cubic ...
Basic equations of interfacial area transport in gas-liquid two-phase flow
International Nuclear Information System (INIS)
The rigorous and consistent formulations of basic equations of interfacial area transport were derived using correlation functions of characteristic function of each phase and velocities of each phase. Turbulent transport term of interfacial area concentration was consistently derived and related to the difference between interfacial velocity and averaged velocity of each phase. Constitutive equations of turbulent transport terms of interfacial area concentration were proposed for bubbly flow. New transport model and constitutive equations were developed for churn flow. These models and constitutive equations are validated by experimental data of radial distributions of interfacial area concentration in bubbly and churn flow. (author)
International Nuclear Information System (INIS)
A new and rigorous modeling of basic transport equation of interfacial area concentration was carried out. Based on the local instant formulation of gas-liquid two-phase flow, basic transport equation of interfacial area concentration was rigorously formulated. In the basic transport equations, the averaged interfacial velocity was shown to be correlation functions of fluctuation of velocity and local instant void fraction and their derivatives which reflect the transport of interfacial area concentration due to interaction between interfacial area and turbulence of each phase. Basic conservation equations of spatial correlation functions of characteristic function and velocity of each phase were also derived based on the conservation equations momentum and its fluctuation of each phase. (author)
Sman, van der R.G.M.
2006-01-01
In the special case of relaxation parameter = 1 lattice Boltzmann schemes for (convection) diffusion and fluid flow are equivalent to finite difference/volume (FD) schemes, and are thus coined finite Boltzmann (FB) schemes. We show that the equivalence is inherent to the homology of the Maxwell-Bolt
Elliptic random-walk equation for suspension and tracer transport in porous media
DEFF Research Database (Denmark)
Shapiro, Alexander; Bedrikovetsky, P. G.
2008-01-01
We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time...
Energy Technology Data Exchange (ETDEWEB)
Chatterjee, Kausik, E-mail: kausik.chatterjee@aggiemail.usu.edu [Strategic and Military Space Division, Space Dynamics Laboratory, North Logan, UT 84341 (United States); Center for Atmospheric and Space Sciences, Utah State University, Logan, UT 84322 (United States); Roadcap, John R., E-mail: john.roadcap@us.af.mil [Air Force Research Laboratory, Kirtland AFB, NM 87117 (United States); Singh, Surendra, E-mail: surendra-singh@utulsa.edu [Department of Electrical Engineering, The University of Tulsa, Tulsa, OK 74104 (United States)
2014-11-01
The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.
International Nuclear Information System (INIS)
The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications
International Nuclear Information System (INIS)
Light transport in graded index media follows a curved trajectory determined by Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.
Crystallographic Lattice Boltzmann Method.
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-06-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows.
Energy Technology Data Exchange (ETDEWEB)
Wilcox, T. P.
1973-09-20
The code ANISN-L solves the one-dimensional, multigroup, time-independent Boltzmann transport equation by the method of discrete ordinates. In problems involving a fissionable system, it can calculate the system multiplication or alpha. In such cases, it is also capable of determining isotopic concentrations, radii, zone widths, or buckling in order to achieve a given multiplication or alpha. The code may also calculate fluxes caused by a specified fixed source. Neutron, gamma, and coupled neutron--gamma problems may be solved in either the forward or adjoint (backward) modes. Cross sections describing upscatter, as well as the usual downscatter, may be employed. This report describes the use of ANISN-L; this is a revised version of ANISN which handles both large and small problems efficiently on CDC-7600 computers. (RWR)
High Order Numerical Solution of Integral Transport Equation in Slab Geometry
Institute of Scientific and Technical Information of China (English)
沈智军; 袁光伟; 沈隆钧
2002-01-01
@@ There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].
Lattice Boltzmann approaches to magnetohydrodynamics and electromagnetism
Dellar, Paul
2010-03-01
J u B E g We present a lattice Boltzmann approach for magnetohydrodynamics and electromagnetism that expresses the magnetic field using a discrete set of vector distribution functions i. The i were first postulated to evolve according to a vector Boltzmann equation of the form ti+ ξi.∇i= - 1τ ( i- i^(0) ), where the ξi are a discrete set of velocities. The right hand side relaxes the i towards some specified functions i^(0) of the fluid velocity , and of the macroscopic magnetic field given by = ∑ii. Slowly varying solutions obey the equations of resistive magnetohydrodynamics. This lattice Boltzmann formulation has been used in large-scale (up to 1800^3 resolution) simulations of magnetohydrodynamic turbulence. However, this is only the simplest form of Ohm's law. We may simulate more realistic extended forms of Ohm's law using more complex collision operators. A current-dependent relaxation time yields a current-dependent resistivity η(|∇x|), as used to model ``anomalous'' resistivity created by small-scale plasma processes. Using a hydrodynamic matrix collision operator that depends upon the magnetic field , we may simulate Braginskii's magnetohydrodynamics, in which the viscosity for strains parallel to the magnetic field lines is much larger than the viscosity for strains in perpendicular directions. Changing the collision operator again, from the above vector Boltzmann equation we may derive the full set of Maxwell's equations, including the displacement current, and Ohm's law, - 1c^2 tE+ ∇x= μo,= σ( E + x). The original lattice Boltzmann scheme was designed to reproduce resistive magnetohydrodynamics in the non-relativistic limit. However, the kinetic formulation requires a system of first order partial differential equations with collision terms. This system coincides with the full set of Maxwell's equations and Ohm's law, so we capture a much wider range of electromagnetic phenomena, including electromagnetic waves.
Wawrzynczak, A; Kluczek, M
2015-01-01
We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.
Transport equation for plasmas in a stationary-homogeneous turbulence
Wang, Shaojie
2015-01-01
For a plasma in a stationary homogeneous turbulence, the Fokker-Planck equation is derived from the nonlinear Vlasov equation by introducing the entropy principle. The ensemble average in evaluating the kinetic diffusion tensor, whose symmetry has been proved, can be computed in a straightforward way when the fluctuating particle trajectories are provided. As an application, it has been shown that a mean electric filed can drive a particle flux through the Stokes-Einstein relation, independent of the details of turbulence.
Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation
Directory of Open Access Journals (Sweden)
J. F. Gómez Aguilar
2016-01-01
Full Text Available The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range (0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to 1.
Kikuchi, Yuta; Tsumura, Kyosuke; Kunihiro, Teiji
2016-01-01
We give a detailed derivation of the second-order (local) hydrodynamics for Boltzmann equation with an external force by using the renormalization group method. In this method, we solve the Boltzmann equation faithfully to extract the hydrodynamics without recourse to any ansatz. Our method leads to microscopic expressions of not only all the transport coefficients that are of the same form as those in Chapman-Enskog method but also those of the viscous relaxation times $\\tau_i$ that admit ph...
Sun, Shuyu
2012-06-02
A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.
Transport methods and interactions for space radiations
Wilson, John W.; Townsend, Lawrence W.; Schimmerling, Walter S.; Khandelwal, Govind S.; Khan, Ferdous S.; Nealy, John E.; Cucinotta, Francis A.; Simonsen, Lisa C.; Shinn, Judy L.; Norbury, John W.
1991-01-01
A review of the program in space radiation protection at the Langley Research Center is given. The relevant Boltzmann equations are given with a discussion of approximation procedures for space applications. The interaction coefficients are related to solution of the many-body Schroedinger equation with nuclear and electromagnetic forces. Various solution techniques are discussed to obtain relevant interaction cross sections with extensive comparison with experiments. Solution techniques for the Boltzmann equations are discussed in detail. Transport computer code validation is discussed through analytical benchmarking, comparison with other codes, comparison with laboratory experiments and measurements in space. Applications to lunar and Mars missions are discussed.
Riaud, Antoine; Zhao, Shufang; Wang, Kai; Cheng, Yi; Luo, Guangsheng
2014-05-01
Despite the popularity of the lattice-Boltzmann method (LBM) in simulating multiphase flows, a general approach for modeling dilute species in multiphase systems is still missing. In this report we propose to modify the collision operator of the solute by introducing a modified redistribution scheme. This operator is based on local fluid variables and keeps the parallelism inherent to LBM. After deriving macroscopic transport equations, an analytical equation of state of the solute is exhibited and the method is proven constituting a unified framework to simulate arbitrary solute distribution between phases, including single-phase soluble compounds, amphiphilic species with a partition coefficient, and surface-adsorbed compounds. PMID:25353915
Transport equation for plasmas in a stationary-homogeneous turbulence
Energy Technology Data Exchange (ETDEWEB)
Wang, Shaojie, E-mail: wangsj@ustc.edu.cn [Department of Modern Physics, University of Science and Technology of China, Hefei 230026 (China)
2016-02-15
For a plasma in a stationary homogeneous turbulence, the Fokker-Planck equation is derived from the nonlinear Vlasov equation by introducing the entropy principle. The ensemble average in evaluating the kinetic diffusion tensor, whose symmetry has been proved, can be computed in a straightforward way when the fluctuating particle trajectories are provided. As an application, it has been shown that a mean parallel electric filed can drive a particle flux through the Stokes-Einstein relation, independent of the details of the fluctuations.
Simple jumping process with memory: Transport equation and diffusion
Kamińska, A.; Srokowski, T.
2004-06-01
We present a stochastic jumping process, defined in terms of jump-size probability density and jumping rate, which is a generalization of the well-known kangaroo process. The definition takes into account two process values: after and before the jump. Therefore, the process is able to preserve memory about its previous values. It possesses a simple stationary limit. Its master equation is interpreted as the kinetic equation with variable collision rate. The process can be easily applied to model systems which relax to distributions other than Maxwellian. The case of a constant jumping rate corresponds to the diffusion process, either normal or ballistic.
Campos-García, Manuel; Granados-Agustín, Fermín.; Cornejo-Rodríguez, Alejandro; Estrada-Molina, Amilcar; Avendaño-Alejo, Maximino; Moreno-Oliva, Víctor Iván.
2013-11-01
In order to obtain a clearer interpretation of the Intensity Transport Equation (ITE), in this work, we propose an algorithm to solve it for some particular wavefronts and its corresponding intensity distributions. By simulating intensity distributions in some planes, the ITE is turns into a Poisson equation with Neumann boundary conditions. The Poisson equation is solved by means of the iterative algorithm SOR (Simultaneous Over-Relaxation).
Role of Dielectric Constant on Ion Transport: Reformulated Arrhenius Equation
Directory of Open Access Journals (Sweden)
Shujahadeen B. Aziz
2016-01-01
Full Text Available Solid and nanocomposite polymer electrolytes based on chitosan have been prepared by solution cast technique. The XRD results reveal the occurrence of complexation between chitosan (CS and the LiTf salt. The deconvolution of the diffractogram of nanocomposite solid polymer electrolytes demonstrates the increase of amorphous domain with increasing alumina content up to 4 wt.%. Further incorporation of alumina nanoparticles (6 to 10 wt.% Al2O3 results in crystallinity increase (large crystallite size. The morphological (SEM and EDX analysis well supported the XRD results. Similar trends of DC conductivity and dielectric constant with Al2O3 concentration were explained. The TEM images were used to explain the phenomena of space charge and blocking effects. The reformulated Arrhenius equation (σ(ε′,T=σoexp(-Ea/KBε′T was proposed from the smooth exponential behavior of DC conductivity versus dielectric constant at different temperatures. The more linear behavior of DC conductivity versus 1000/(ɛ′×T reveals the crucial role of dielectric constant in Arrhenius equation. The drawbacks of Arrhenius equation can be understood from the less linear behavior of DC conductivity versus 1000/T. The relaxation processes have been interpreted in terms of Argand plots.
Institute of Scientific and Technical Information of China (English)
Ying Genjun; Fu Ying; Ma Yichen; Zhang Zhipeng
2006-01-01
We present an algorithm for numerical solution of transport equation in diffusive regimes, in which the transport equation is nearly singular and its solution becomes a solution of a diffusion equation. This algorithm, which is based on the Least-squares FEM in combination with a scaling transformation, presents a good approximation of a diffusion operator in diffusive regimes and guarantees an accurate discrete solution. The numerical experiments in 2D and 3D case are given, and the numerical results show that this algorithm is correct and efficient.
The adjoint neutron transport equation and the statistical approach for its solution
Saracco, Paolo; Ravetto, Piero
2016-01-01
The adjoint equation was introduced in the early days of neutron transport and its solution, the neutron importance, has ben used for several applications in neutronics. The work presents at first a critical review of the adjoint neutron transport equation. Afterwards, the adjont model is constructed for a reference physical situation, for which an analytical approach is viable, i.e. an infinite homogeneous scattering medium. This problem leads to an equation that is the adjoint of the slowing-down equation that is well-known in nuclear reactor physics. A general closed-form analytical solution to such adjoint equation is obtained by a procedure that can be used also to derive the classical Placzek functions. This solution constitutes a benchmark for any statistical or numerical approach to the adjoint equation. A sampling technique to evaluate the adjoint flux for the transport equation is then proposed and physically interpreted as a transport model for pseudo-particles. This can be done by introducing appr...
Least-squares finite element discretizations of neutron transport equations in 3 dimensions
Energy Technology Data Exchange (ETDEWEB)
Manteuffel, T.A [Univ. of Colorado, Boulder, CO (United States); Ressel, K.J. [Interdisciplinary Project Center for Supercomputing, Zurich (Switzerland); Starkes, G. [Universtaet Karlsruhe (Germany)
1996-12-31
The least-squares finite element framework to the neutron transport equation introduced in is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. Here we report on some practical aspects of this approach for neutron transport calculations in three space dimensions. The systems of partial differential equations resulting from a P{sub 1} and P{sub 2} approximation of the angular dependence are derived. In the diffusive limit, the system is essentially a Poisson equation for zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. We report on the numerical results obtained for the minimum of the least-squares functional augmented by an additional boundary term using trilinear finite elements on a uniform tesselation into cubes.
A numerical method for the elliptic Monge-Amp\\`ere equation with transport boundary conditions
Froese, Brittany D
2011-01-01
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\\`ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary conditions. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Amp\\`ere equation with Neumann boundary conditions. We present a new discretization for the equation, which converges to the viscosity solution. The resulting system is solved efficiently with Newton's method. We provide several challenging computational examples that demonstrate the effectiveness and efficiency ($O(M)-O(M^{1.3})$ time) of the proposed method.
Electron transport data in N2-O2 streamer plasma discharges
Dujko, S.; Ebert, U.; White, R.D.; Petrović, Z.L.
2010-01-01
A multi-term theory for solving the Boltzmann equation and a Monte Carlo simulation technique are used to investigate the electron transport in mixtures of molecular nitrogen and oxygen. We investigate the way in which the transport coefficients and spatially resolved transport data are influenced b
White, G A
2015-01-01
We propose a general method to analytically solve transport equations during a cosmic phase transition without making approximations based on the assumption that any transport coefficient is large. Using the MSSM as an example we derive the solutions to a set of $3$ transport equations derived under the assumption of supergauge equilibrium and the diffusion approximation. The result is then rederived efficiently using a technique we present involving a parametrized ansatz which turns the process of deriving a solution into an almost elementary problem. We then show how both the derivation and the parametrized ansatz technique can be generalized to solve an arbitrary number of transport equations. Finally we derive a perturbative series that relaxes the usual approximation that inactivates VEV dependent relaxation and CP violating source terms at the bubble wall and through the symmetric phase.
General analytic methods for solving coupled transport equations: From cosmology to beyond
White, G. A.
2016-02-01
We propose a general method to analytically solve transport equations during a phase transition without making approximations based on the assumption that any transport coefficient is large. Using a cosmic phase transition in the minimal supersymmetric standard model as a pedagogical example, we derive the solutions to a set of 3 transport equations derived under the assumption of supergauge equilibrium and the diffusion approximation. The result is then rederived efficiently using a technique we present involving a parametrized ansatz which turns the process of deriving a solution into an almost elementary problem. We then show how both the derivation and the parametrized ansatz technique can be generalized to solve an arbitrary number of transport equations. Finally we derive a perturbative series that relaxes the usual approximation that inactivates vacuum-expectation-value dependent relaxation and C P -violating source terms at the bubble wall and through the symmetric phase. Our analytical methods are able to reproduce a numerical calculation in the literature.
Institute of Scientific and Technical Information of China (English)
2008-01-01
A discrete ordinates method for a threedimensional first-order neutron transport equation based on unstructured-meshes that avoids the singularity of the second-order neutron transport equation in void regions was derived.The finite element variation equation was obtained using the least-squares method.A three-dimensional transport calculation code was developed.Both the triangular-z and the tetrahedron elements were included.The numerical results of some benchmark problems demonstrated that this method can solve neutron transport problems in unstructuredmeshes very well.For most problems,the error of the eigenvalue and the angular flux is less than 0.3% and 3.0% respectively.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
The isotropic diffusion source approximation for supernova neutrino transport
Liebendoerfer, M.; Whitehouse, S. C.; Fischer, T
2007-01-01
Astrophysical observations originate from matter that interacts with radiation or transported particles. We develop a pragmatic approximation in order to enable multi-dimensional simulations with basic spectral radiative transfer when the computational resources are not sufficient to solve the complete Boltzmann transport equation. The distribution function of the transported particles is decomposed into trapped and streaming particle components. Their separate evolution equations are coupled...
The suspended sediment transport equation and its near-bed sediment flux
Institute of Scientific and Technical Information of China (English)
LI RuiJie; LUO Feng; ZHU WenJin
2009-01-01
The suspended sediment transport equation and its near-bed sediment flux are one of the key prob-lems of sediment transport research under nonequilibrium condition. Based on the three-dimensional primitive suspended transport equation, the two-dimensional suspended sediment transport equation is deduced. The derived process indicates that the physical essence of the near-bed sediment flux is right the bottom boundary condition for the suspended sediment transport equation. This paper ana-lyzes the internal relations between the two methods of sediment carrying capacity and shear stress in common use, points out the consistency of these two methods in terms of form and physical meaning, and unifies these two methods theoretically. Furthermore, based on the analysis and comparison of the expressions of the near-bed sediment flux, this paper summarizes some problems to which attention should be paid, thus offering a novel approach to the study and the solution of the problems of sus-pended sediment transport and exchange flux of near-bed water sediment.
The suspended sediment transport equation and its near-bed sediment flux
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The suspended sediment transport equation and its near-bed sediment flux are one of the key problems of sediment transport research under nonequilibrium condition. Based on the three-dimensional primitive suspended transport equation, the two-dimensional suspended sediment transport equation is deduced. The derived process indicates that the physical essence of the near-bed sediment flux is right the bottom boundary condition for the suspended sediment transport equation. This paper analyzes the internal relations between the two methods of sediment carrying capacity and shear stress in common use, points out the consistency of these two methods in terms of form and physical meaning, and unifies these two methods theoretically. Furthermore, based on the analysis and comparison of the expressions of the near-bed sediment flux, this paper summarizes some problems to which attention should be paid, thus offering a novel approach to the study and the solution of the problems of suspended sediment transport and exchange flux of near-bed water sediment.
Training Restricted Boltzmann Machines
DEFF Research Database (Denmark)
Fischer, Asja
Restricted Boltzmann machines (RBMs) are probabilistic graphical models that can also be interpreted as stochastic neural networks. Training RBMs is known to be challenging. Computing the likelihood of the model parameters or its gradient is in general computationally intensive. Thus, training...
The H sub N method for solving linear transport equation: theory and applications
Tezcan, C; Guelecyuez, M C
2003-01-01
The system of singular integral equations which is obtained from the integro-differential form of the linear transport equation using the Placzek lemma is solved. The exit distributions at the boundaries of the various media and the infinite medium Green's function are used. The process is applied to the half-space and finite slab problems. The neutron angular density in terms of singular eigenfunctions of the method of elementary solutions is also used to derive the same analytical expressions.
Transient 1D transport equation simulated by a mixed Green element formulation
Taigbenu, Akpofure Efemena; Onyejekwe, Okey Oseloka
1997-08-01
New discrete element equations or coefficients are derived for the transient 1D diffusion-advection or transport equation based on the Green element replication of the differential equation using linear elements. The Green element method (GEM), which solves the singular boundary integral theory (a Fredholm integral equation of the second kind) on a typical element, gives rise to a banded global coefficient matrix which is amenable to efficient matrix solvers. It is herein derived for the transient 1D transport equation with uniform and non-uniform ambient flow conditions and in which first-order decay of the containment is allowed to take place. Because the GEM implements the singular boundary integral theory within each element at a time, the integrations are carried out in exact fashion, thereby making the application of the boundary integral theory more utilitarian. This system of discrete equations, presented herein for the first time, using linear interpolating functions in the spatial dimensions shows promising stable characteristics for advection-dominant transport. Three numerical examples are used to demonstrate the capabilities of the method. The second-order-correct Crank-Nicolson scheme and the modified fully implicit scheme with a difference weighting value of two give superior solutions in all simulated examples.
Kinetic theory the Chapman-Enskog solution of the transport equation for moderately dense gases
Brush, S G
1972-01-01
Kinetic Theory, Volume 3: The Chapman-Enskog Solution of the Transport Equation for Moderately Dense Gases describes the Chapman-Enskog solution of the transport equation for moderately dense gases. Topics covered range from the propagation of sound in monatomic gases to the kinetic theory of simple and composite monatomic gases and generalizations of the theory to higher densities. The application of kinetic theory to the determination of intermolecular forces is also discussed. This volume is divided into two sections and begins with an introduction to the work of Hilbert, Chapman, and Ensko
Review of characteristic methods used to solve the linear transport equation
International Nuclear Information System (INIS)
Characteristic methods used to solve the linear transport equation are reviewed. Characteristic methods are based upon the solution of the transport equation written in the form psi(s) = psi(s0)e/sup -sigma/sub T/(s-s0)/ + ∫/sub s0//sup s/Q(t')e/sup -sigma/sub T/(s-t')/dt', where s is arc length along the characteristic. The methods of solution distinguish themselves in how the characteristics used for computation are selected and how the source term is approximated. Criteria upon which a production method should be based are recommended. 3 figures
The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit
Energy Technology Data Exchange (ETDEWEB)
Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-07-12
We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.
Solving the transport equation with quadratic finite elements: Theory and applications
Energy Technology Data Exchange (ETDEWEB)
Ferguson, J.M. [Lawrence Livermore National Lab., CA (United States)
1997-12-31
At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids.
The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit
Energy Technology Data Exchange (ETDEWEB)
Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-07-12
We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radi- ation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material tem- peratures are always di erent in time dependent problems, and we use it to show that IMC does not produce the correct di usion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.
Directory of Open Access Journals (Sweden)
Shulin Wu
2009-01-01
Full Text Available We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.
New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory
Sakthivel, Rathinasamy; Chun, Changbum; Lee, Jonu
2010-09-01
The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation
Relativistic quantum transport coefficients for second-order viscous hydrodynamics
Florkowski, Wojciech; Maksymiuk, Ewa; Ryblewski, Radoslaw; Strickland, Michael
2015-01-01
We express the transport coefficients appearing in the second-order evolution equations for bulk viscous pressure and shear stress tensor using Bose-Einstein, Boltzmann, and Fermi-Dirac statistics for the equilibrium distribution function and Grad's 14-moment approximation as well as the method of Chapman-Enskog expansion for the non-equilibrium part. Specializing to the case of boost-invariant and transversally homogeneous longitudinal expansion of the viscous medium, we compare the results obtained using the above methods with those obtained from the exact solution of massive 0+1d Boltzmann equation in the relaxation-time approximation. We show that compared to the 14-moment approximation, the hydrodynamic transport coefficients obtained using the Chapman-Enskog method result in better agreement with the exact solution of the Boltzmann equation in relaxation-time approximation.
Classical Theory of Hot-Electron Transport in Electric and Magnetic Fields
Institute of Scientific and Technical Information of China (English)
WENG Ming-Qi; WU Hang-Sheng
2002-01-01
Balance equation approach to the hot-electron transport in electric and magnetic fields is reformulated.The balance equations are re-derived from the Boltzmann equation. A new expression for the distribution function isreported in the present paper. It is homogeneous steady solution of the Boltzmann equation in constant relaxation timeapproximation. It holds when ωocτ < i or ωc < Te. As an example, the mobility of 2D electron gas in the GaAs-AlGaAsheterojunction is computed as a function of electric field and magnetic field.
Classical Theory of Hot－Electron Transport in Electric and Magnetic Fields
Institute of Scientific and Technical Information of China (English)
WENGMing－Qi; WuHang－Sheng
2002-01-01
Balance equation approach to the hot-electron transport in electric and magnetic fields is reformulated.The balance equations are re-derived from the Boltzmann equation.A new expression for the distribution function is reported in the present paper.It is homogeneous steady solution of the Boltzmann equation in costant relaxation time approximation.It holds when ωT<<1 or ωc<
L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
International Nuclear Information System (INIS)
We present a L2-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L2-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L2-stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L2-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L2-stability estimate. This is the first result on the L2-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions
A Derivation of the Nonlocal Volume-Averaged Equations for Two-Phase Flow Transport
Directory of Open Access Journals (Sweden)
Gilberto Espinosa-Paredes
2012-01-01
Full Text Available In this paper a detailed derivation of the general transport equations for two-phase systems using a method based on nonlocal volume averaging is presented. The local volume averaging equations are commonly applied in nuclear reactor system for optimal design and safe operation. Unfortunately, these equations are limited to length-scale restriction and according with the theory of the averaging volume method, these fail in transition of the flow patterns and boundaries between two-phase flow and solid, which produce rapid changes in the physical properties and void fraction. The non-local volume averaging equations derived in this work contain new terms related with non-local transport effects due to accumulation, convection diffusion and transport properties for two-phase flow; for instance, they can be applied in the boundary between a two-phase flow and a solid phase, or in the boundary of the transition region of two-phase flows where the local volume averaging equations fail.
Measuring the contour of a wavefront using the Irradiance Transport Equation (ITE)
Castillo-Rodríguez, Luis; Granados-Agustín, Fermín; Fernández-Guasti, Manuel; Cornejo-Rodríguez, Alejandro
2006-01-01
The Irradiance Transport Equation (ITE), found by Teague, had been used in optics with different applications. One of the field where had been used is in optical testing, for example, with the method developed by Takeda. In this paper following the idea of using different optical and mathematical analysis method, theorical and experimental results are presented.
Bal, Guillaume; Schotland, John C
2015-01-01
We propose a method to reconstruct the density of an optical source in a highly scattering medium from ultrasound-modulated optical measurements. Our approach is based on the solution to a hybrid inverse source problem for the radiative transport equation (RTE). A controllability result for the RTE plays an essential role in the analysis.
Stochastic approach to the numerical solution of the non-stationary Parker's transport equation
Wawrzynczak, A; Gil, A
2015-01-01
We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The obtained stochastic model of the Forbu...
Analytical solutions of the advection-dispersion equation and related models are indispensable for predicting or analyzing contaminant transport processes in streams and rivers, as well as in other surface water bodies. Many useful analytical solutions originated in disciplines other than surface-w...
Analytical solution for the advection-dispersion transport equation in layered media
The advection-dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection-diffusion eigenvalue problem that had the same form and coef...
Nuclear Flow in Consistent Boltzmann Algorithm Models
Kortemeyer, G.; Daffin, F.; Bauer, W.
1995-01-01
We investigate the stochastic Direct Simulation Monte Carlo method (DSMC) for numerically solving the collision-term in heavy-ion transport theories of the Boltzmann-Uehling-Uhlenbeck (BUU) type. The first major modification we consider is changes in the collision rates due to excluded volume and shadowing/screening effects (Enskog theory). The second effect studied by us is the inclusion of an additional advection term. These modifications ensure a non-vanishing second virial and change the ...
Energy Technology Data Exchange (ETDEWEB)
Jin, Shi, E-mail: jin@math.wisc.edu [Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240 (China); Department of Mathematics, University of Wisconsin, Madison, WI 53706 (United States); Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu [Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (United States); Department of Mathematics, University of Utah, Salt Lake City, UT 84112 (United States); Zhu, Xueyu, E-mail: xzhu@sci.utah.edu [Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (United States)
2015-05-15
In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.
Number-resolved master equation approach to quantum measurement and quantum transport
Li, Xin-Qi
2016-08-01
In addition to the well-known Landauer-Büttiker scattering theory and the nonequilibrium Green's function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number ( n)-resolved master equation ( n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.
International Nuclear Information System (INIS)
An additive angular-dependent re-balance (AADR) factor acceleration method is described to accelerate the source iteration of discrete ordinates transport calculation. The formulation of the AADR method follows that of the angular-dependent re-balance (ADR) method in that the re-balance factor is defined only on the cell interface and in that the low-order equation is derived by integrating the transport equation (high-order equation) over angular subspaces. But, the re-balance factor is applied additively. While the AADR method is similar to the boundary projection acceleration and the alpha-weighted linear acceleration, it is more general and does have distinct features. The method is easily extendible to DPN and low-order SN re-balancing, and it does not require consistent discretizations between the high- and low-order equations as in diffusion synthetic acceleration. We find by Fourier analysis and numerical results that the AADR method with a chosen form of weighting functions is unconditionally stable and very effective. There also exists an optimal weighting parameter that leads to the smallest spectral radius. The AADR acceleration method described in this paper is simple to implement, unconditionally stable, and very effective. It uses a physically based weighting function with an optimal parameter, leading to the best spectral radius of ρ<0.1865, compared to ρ<0.2247 of DSA. The application of the AADR acceleration method with the LMB scheme on a test problem shows encouraging results
Homogenization of the Poisson-Nernst-Planck Equations for Ion Transport in Charged Porous Media
Schmuck, Markus
2012-01-01
Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media. Homogenization analysis is performed for a two-component pe- riodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Three new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to mean pore size; and (iii) the surface charge per volume appears as a continuous "background charge density". The coeffcient tensors in the macroscopic PNP equations can be calculated from periodic reference cell problem, and several examples are considered. For an insulating solid matrix, all gradients are corrected by a single tortuosit...
Degond, Pierre; Juengel, Ansgar; Pietra, Paola
1999-01-01
The energy-transport models describe the flow of electrons through a semiconductor crystal, influenced by diffusive, electrical and thermal effects. They consist of the continuity equations for the mass and the energy, coupled to Poisson's equation for the electric potential. These models can be derived from the semiconductor Boltzmann equation.This paper consists of two parts. The first part concerns with the modelling of the energy-transport system. The diffusion coefficients and the energy...
On Boltzmann's genius and thermodynamics
Gyftopoulos, Elias P
2007-01-01
A recent essay [1] reminds us of how richly Boltzmann deserves to be admiringly commemorated for the originality of his ideas on the occasion of his 150th birthday. Without any doubt, the scientific community owes Boltzmann a great debt of gratitude for his ingenious and pathfinding contributions. However, the essay chooses to illustrate this important memorial by statements and inferences that perhaps are questionable today even to Boltzmann himself. I will comment only on three issues.
Energy Technology Data Exchange (ETDEWEB)
Morel, J.E.
1981-01-01
A collocation method is developed for the solution of the one-dimensional neutron transport equation in slab geometry with both symmetric and polarly asymmetric scattering. For the symmetric scattering case, it is found that the collocation method offers a combination of some of the best characteristics of the finite-element and discrete-ordinates methods. For the asymmetric scattering case, it is found that the computational cost of cross-section data processing under the collocation approach can be significantly less than that associated with the discrete-ordinates approach. A general diffusion equation treating both symmetric and asymmetric scattering is developed and used in a synthetic acceleration algorithm to accelerate the iterative convergence of collocation solutions. It is shown that a certain type of asymmetric scattering can radically alter the asymptotic behavior of the transport solution and is mathematically equivalent within the diffusion approximation to particle transport under the influence of an electric field. The method is easily extended to other geometries and higher dimensions. Applications exist in the areas of neutron transport with highly anisotropic scattering (such as that associated with hydrogenous media), charged-particle transport, and particle transport in controlled-fusion plasmas. 23 figures, 6 tables.
Institute of Scientific and Technical Information of China (English)
李志辉; 彭傲平; 方方; 李四新; 张顺玉
2015-01-01
How to solve hypersonic aerothermodynamics and complex flow mechanism covering various flow regimes from high rarefied free-molecular flow of outer-layer space to continuum flow of near-ground is one of the frontier basic problems in the field of fluid physics. In this work, the unified Boltzmann model equation based on the molecular velocity distribution function is presented for describing complex hypersonic flow transport phenomena covering all flow regimes by physics analysis and model processing of the collision integral to the Boltzmann equation. The discrete velocity ordinate method is developed to simulate complex flows from low Mach numbers to hypersonic flight, and the gas-kinetic coupling-iteration numerical scheme is constructed directly to solve the evolution and updating of the molecular velocity distribution function by employing the unsteady time-splitting method and the NND finite-difference technique. Then, the gas-kinetic unified algorithm (GKUA) is presented to simulate the three-dimensional hypersonic aerothermodynamics and flow problems around space vehicles covering various flow regimes from free-molecule to continuum. To verify the accuracy and reliability of the present GKUA and simulate gas thermodynamic transport phenomena covering various flow regimes, firstly, the two-dimensional supersonic flows around a circular cylinder are simulated in the continuum regime of Kn∞ = 0.0001 and in the high rarefied regime of Kn∞ = 0.3 through the comparison between the Navier-Stokes (N-S) solution and the direct simulation Monte Carlo (DSMC) result, respectively. It is indicated that the GKUA can exactly converge to the N-S solution in the continuum flow regime, and the computed results of the GKUA are consistent with the DSMC simulation with a small deviation of 0.45%in the high rarefied flow regime. Then, the three-dimensional complex hypersonic flows around reusable satellite shape are studied as one of the engineering applications of the GKUA
Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond
Energy Technology Data Exchange (ETDEWEB)
Stockamp, T.
2006-12-22
In chapter 2 we chose the real time formalism to discuss some basic principles in quantum field theory at finite temperature. This enables us to derive the quantum Boltzmann equation from the Schwinger-Dyson series. We then shortly introduce the basic concepts of QCD which are needed to understand the physics of QGP formation. After a detailed account on the bottom-up scenario we show the consistency of this approach by a diagramatical analysis of the relevant Boltzmann collision integrals. Chapter 3 deals with BEC dynamics out of equilibrium. After an introduction to the fundamental theoretical tool - namely the Gross-Pitaevskii equation - we focus on a generalization to finite temperature developed by Zaremba, Nikuni and Griffin (ZNG). These authors use a Boltzmann equation to describe the interactions between condensed and excited atoms and manage in this way to describe condensate growth. We then turn to a discussion on the 2PI effective action and derive equations of motion for a relativistic scalar field theory. In the nonrelativistic limit these equations are shown to coincide with the ZNG theory when a quasiparticle approximation is applied. Finally, we perform a numerical analysis of the full 2PI equations. These remain valid even at strong coupling and far from equilibrium, and thus go far beyond Boltzmann's approach. For simplicity, we limit ourselves to a homogeneous system and present the first 3+1 dimensional study of condensate melting. (orig.)
Note on the Solution of Transport Equation by Tau Method and Walsh Functions
Directory of Open Access Journals (Sweden)
Abdelouahab Kadem
2010-01-01
Full Text Available We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice ...
Lattice Boltzmann method and its applications in engineering thermophysics
Institute of Scientific and Technical Information of China (English)
HE YaLing; LI Qing; WANG Yong; TANG GuiHua
2009-01-01
The lattice Boltzmann method (LBM),a mesoscopic method between the molecular dynamics method and the conventional numerical methods,has been developed into a very efficient numerical alternative in the past two decades.Unlike conventional numerical methods,the kinetic theory based LBM simulates fluid flows by tracking the evolution of the particle distribution function,and then accumulates the distribution to obtain macroscopic averaged properties.In this article we review some work on LBM applications in engineering thermophysics:(1) brief introduction to the development of the LBM; (2)fundamental theory of LBM including the Boltzmann equation,Maxwell distribution function,Boltzmann-BGK equation,and the lattice Boltzmann-BGK equation; (3) lattice Boltzmann models for compressible flows and non-equilibrium gas flows,bounce back-specular-reflection boundary scheme for microscale gaseous flows,the mass modified outlet boundary scheme for fully developed flows,and an implicit-explicit finite-difference-based LBM; and (4) applications of the LBM to oscillating flow,compressible flow,porous media flow,non-equilibrium flow,and gas resonant oscillating flow.
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
Degond, Pierre; Liu, Hailiang; Savelief, Dominique; Vignal, Marie-Hélène
2012-01-01
International audience This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare...
Daiki Tanaka; Nobuyuki Higashimori; Yuusuke Iso
2012-01-01
The aim of the paper is to give $L^p$-estimates for the weak solution to the transport equation with $L^\\infty$-perturbation both of the attenuation coefficient and of the scattering kernel. We try to clarify the constants in our estimates, and we show the estimates by a direct calculation. We introduce the albedo operator for the solutions, and we show stability of perturbed albedo operators.
Directory of Open Access Journals (Sweden)
Daiki Tanaka
2012-06-01
Full Text Available The aim of the paper is to give $L^p$-estimates for the weak solution to the transport equation with $L^\\infty$-perturbation both of the attenuation coefficient and of the scattering kernel. We try to clarify the constants in our estimates, and we show the estimates by a direct calculation. We introduce the albedo operator for the solutions, and we show stability of perturbed albedo operators.
Christlieb, Andrew J.; Liu, Yuan; Tang, Qi; Xu, Zhengfu
2014-01-01
In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition and positivity of...
Numerical solution of neutron transport equations in discrete ordinates and slab geometry
International Nuclear Information System (INIS)
An unified formalism to solve numerically, between other equation, the neutron transport in discrete ordinates, slab geometry, several energy groups and independents of time, has been developed recently. Such a formalism cover some of the conventional schemes as diamond difference, (WDD) characteristic step (SC) lineal characteristic (LC), quadratic characteristic (QC) and lineal discontinuous. Unified formation gives before hand the convergence order of the previously selected scheme. In fact it allows besides to generate a big amount of numerical schemes, with which is also possible to solve numerical equations as soon as neutron transport. The essential purpose of this work was to solve the neutron transport equations in slab geometry and discrete ordinates considering several energy groups without to take under advisement time dependence based in the above mentioned unified formalism. To reach this purpose it was necesary to design a computer code with the name TNOD1 (Neutron transport in discrete ordinates and 1 dimension) which includes each one of the schemes already pointed out. there exist two numerical schemes, also recently developed, quadratic continuous (QC) and cubic continuous (CN), although covered by unified formalism, it has been possible to include them inside this computer code without make substantial changes in its structure. In chapter I, derivative of neutron transport equation independent of time is taken, for angular flux, including boundary conditions and discontinuity. In chapter II the neutron transport equations are obtained in multigroups, independents of time, for approximation of discrete ordinates. Description of theory related with unified formalism and its relationship with mentioned discretization schemes is presented in chapter III. Chapter IV describes the computer code developed and finally, in chapter V different numerical results obtained with TNOD1 program are shown. In Appendix A theorems and mathematical arguments used
A stable scheme for computation of coupled transport and equilibrium equations in tokamaks
Fable, E.; Angioni, C.; Ivanov, A. A.; Lackner, K.; Maj, O.; Yu, S.; Medvedev; Pautasso, G.; Pereverzev, G. V.
2013-03-01
The coupled system consisting of 1D radial transport equations and the quasi-static 2D magnetic equilibrium equation for axisymmetric systems (tokamaks) is known to be prone to numerical instabilities, either due to propagation of numerical errors in the iteration process, or due to the choice of the numerical scheme itself. In this paper, a possible origin of these instabilities, specifically associated with the latter condition, is discussed and an approach is chosen, which is shown to have good accuracy and stability properties. This scheme is proposed to be used within those codes for which the poloidal flux ψ is the quantity solved for in the current diffusion equation. Mathematical arguments are used to study the convergence properties of the proposed scheme.
Anderson-Witting transport coefficients for flows in general relativity
Ambrus, Victor E
2016-01-01
The transport coefficients induced by the Anderson-Witting approximation of the collision term in the relativistic Boltzmann equation are derived for close to equilibrium flows in general relativity. Using the tetrad formalism, it is shown that the expression for these coefficients is the same as that obtained on flat space-time, in agreement with the generalized equivalence principle.
Lattice Boltzmann Model for Compressible Fluid on a Square Lattice
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
A two-level four-direction lattice Boltzmann model is formulated on a square lattice to simulate compressible flows with a high Mach number. The particle velocities are adaptive to the mean velocity and internal energy. Therefore, the mean flow can have a high Mach number. Due to the simple form of the equilibrium distribution, the 4th order velocity tensors are not involved in the calculations. Unlike the standard lattice Boltzmann model, o special treatment is need for the homogeneity of 4th order velocity tensors on square lattices. The Navier-Stokes equations were derived by the Chapman-Enskog method from the BGK Boltzmann equation. The model can be easily extended to three-dimensional cubic lattices. Two-dimensional shock-wave propagation was simulated
Accounting for adsorption and desorption in Lattice Boltzmann simulations
Levesque, Maximilien; Pagonabarraga, Ignacio; Frenkel, Daan; Rotenberg, Benjamin
2013-01-01
We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. Associated to the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g. in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic, but also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a Lattice-Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is ...
An integrable 3D lattice model with positive Boltzmann weights
Mangazeev, Vladimir V; Sergeev, Sergey M
2013-01-01
In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0Boltzmann weights.
Analytical Solutions of a Fractional Diffusion-advection Equation for Solar Cosmic-Ray Transport
Litvinenko, Yuri E.; Effenberger, Frederic
2014-12-01
Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.
Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport
Litvinenko, Yuri E
2014-01-01
Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we solve analytically a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, Anna; Jarecka, Dorota; Pawlowska, Hanna; Smolarkiewicz, Piotr K; Waruszewski, Maciej
2014-01-01
This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitati...
Institute of Scientific and Technical Information of China (English)
HUANG Guanhua; HUANG Quanzhong; ZHAN Hongbin; CHEN Jing; XIONG Yunwu; FENG Shaoyuan
2005-01-01
The newly developed Fractional Advection-Dispersion Equation (FADE), which is FADE was extended and used in this paper for modelling adsorbing contaminant transport by adding an adsorbing term. A parameter estimation method and its corresponding FORTRAN based program named FADEMain were developed on the basis of Nonlinear Least Square Algorithm and the analytical solution for one-dimensional FADE under the conditions of step input and steady state flow. Data sets of adsorbing contaminants Cd and NH4+-N transport in short homogeneous soil columns and conservative solute NaCI transport in a long homogeneous soil column, respectively were used to estimate the transport parameters both by FADEMain and the advection-dispersion equation (ADE) based program CXTFIT2.1. Results indicated that the concentration simulated by FADE agreed well with the measured data. Compared to the ADE model, FADE can provide better simulation for the concentration in the initial lower concentration part and the late higher concentration part of the breakthrough curves for both adsorbing contaminants. The dispersion coefficients for ADE were from 0.13 to 7.06 cm2/min, while the dispersion coefficients for FADE ranged from 0.119 to 3.05 cm1.856/min for NaCI transport in the long homogeneous soil column. We found that the dispersion coefficient of FADE increased with the transport distance, and the relationship between them can be quantified with an exponential function. Less scale-dependent was also found for the dispersion coefficient of FADE with respect to ADE.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change.
Kamali, M R; Gillissen, J J J; van den Akker, H E A; Sundaresan, Sankaran
2013-09-01
A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum limit one recovers the well known macroscopic energy conservation equation for the mixtures. Heats of reaction, the enthalpy change associated with the phase change, and the diffusive transport of enthalpy are all taken into account; but the dependence of enthalpy on pressure, which is usually a small effect in most nonisothermal flows encountered in chemical reaction systems, is ignored. The energy equation is coupled to the LB equations for species transport and pseudopotential interaction forces through the EOS by using the filtered local pseudotemperature field. The proposed scheme is validated against simple test problems for which analytical solutions can readily be obtained.
Charge Transport in Weyl Semimetals
Hosur, Pavan; Parameswaran, S. A.; Vishwanath, Ashvin
2011-01-01
We study transport in three dimensional Weyl semimetals with N isotropic Weyl nodes in the presence of Coulomb interactions or disorder at temperature T. In the interacting clean limit, we determine the conductivity by solving a quantum Boltzmann equation within a `leading log' approximation and find it to be proportional to T, upto logarithmic factors arising from the flow of couplings. In the noninteracting disordered case, we compute the finite-frequency Kubo conductivity and show that it ...
Ashworth, D. G.; Bowyer, M. D. J.; Oven, R.
1995-06-01
Commencing with the LSS integro-differential equation, an approximate transport equation is derived from which the moments of the range distribution may be obtained. The resulting equation set is known as the Kent Range ALgorithm (KRAL). The method for numerical solution of these equations, when written as a set of coupled second order ordinary differential equations (ODEs) of the initial value type, is then outlined. Solution is achieved by recasting the equation set in the form of first order ODEs designed for iterative solution. The technique used is an iterative refinement (or residual correction) procedure and the set of first order ODEs is called the Kent Optimised Range ALgorithm (KORAL). Finally, the first three moments from KORAL, first and second order PRAL codes and the full transport equation code KUBBIC-91 are compared with Monte Carlo data obtained from a TRIM code modified to treat targets of infinite extent. Comparisons are performed using consistent nuclear and electronic energy loss models.
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems. PMID:26986435
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice Boltzmann models are applied to simulating the shock tube of one dimension. Good agreement between the numerical results and the analytical solutions are obtained.
Usang, M. D.; Ivanyuk, F. A.; Ishizuka, C.; Chiba, S.
2016-10-01
Nuclear fission is treated by using the Langevin dynamical description with macroscopic and microscopic transport coefficients (mass and friction tensors), and it is elucidated how the microscopic (shell and pairing) effects in the transport coefficients, especially their dependence on temperature, affects various fission observables. We found that the microscopic transport coefficients, calculated by linear response theory, change drastically as a function of temperature: in general, the friction increases with growing temperature while the mass tensor decreases. This temperature dependence brings a noticeable change in the mass distribution and kinetic energies of fission fragments from nuclei around 236U at an excitation energy of 20 MeV. The prescission kinetic energy decreases from 25 MeV at low temperature to about 2.5 MeV at high temperature. In contrast, the Coulomb kinetic energy increases as the temperature increases. Interpolating the microscopic transport coefficients among the various temperatures enabled our Langevin equation to use the microscopic transport coefficients at a deformation-dependent local temperature of the dynamical evolution. This allowed us to compare directly the fission observables of both macroscopic and microscopic calculations, and we found almost identical results under the conditions considered in this work.
BISON, 1-D Burnup and Transport in Slab, Cylindrical, Spherical Geometry
International Nuclear Information System (INIS)
1 - Description of problem or function: BISON-1.5 solves the one- dimensional Boltzmann transport equation for neutron and gamma-rays and transmutation equations for fuel nuclides. 2 - Method of solution: In the transport calculation stage the one- dimensional Boltzmann transport equation is solved by the discrete ordinates method. In the burnup calculation stage, transmutation equations for fuel nuclides are solved by Bateman's method. The neutron flux obtained in the transport calculation stage is used to determine the transmutation rates in the burnup calculation stage. Both stages are repeated in tandem till the end of the burnup cycle. 3 - Restrictions on the complexity of the problem: A 42-group neutron and 21-group gamma-ray cross section library is prepared in the code package. Core storage for array variables is dynamically allocated by the code, so there are no restrictions on the size of each array
The use of Galerkin finite-element methods to solve mass-transport equations
Grove, David B.
1977-01-01
The partial differential equation that describes the transport and reaction of chemical solutes in porous media was solved using the Galerkin finite-element technique. These finite elements were superimposed over finite-difference cells used to solve the flow equation. Both convection and flow due to hydraulic dispersion were considered. Linear and Hermite cubic approximations (basis functions) provided satisfactory results: however, the linear functions were computationally more efficient for two-dimensional problems. Successive over relaxation (SOR) and iteration techniques using Tchebyschef polynomials were used to solve the sparce matrices generated using the linear and Hermite cubic functions, respectively. Comparisons of the finite-element methods to the finite-difference methods, and to analytical results, indicated that a high degree of accuracy may be obtained using the method outlined. The technique was applied to a field problem involving an aquifer contaminated with chloride, tritium, and strontium-90. (Woodard-USGS)
The solution of the multigroup neutron transport equation using spherical harmonics
International Nuclear Information System (INIS)
A solution of the multi-group neutron transport equation in up to three space dimensions is presented. The flux is expanded in a series of unnormalised spherical harmonics. Using the various recurrence formulae a linked set of first order differential equations is obtained for the moments psisup(g)sub(lm)(r), γsup(g)sub(lm)(r). Terms with odd l are eliminated resulting in a second order system which is solved by two methods. The first is a finite difference formulation using an iterative procedure, secondly, in XYZ and XY geometry a finite element solution is given. Results for a test problem using both methods are exhibited and compared. (orig./RW)
Directory of Open Access Journals (Sweden)
Thereza A. Soares
2004-08-01
Full Text Available The ability of biomolecules to catalyze chemical reactions is due chiefly to their sensitivity to variations of the pH in the surrounding environment. The reason for this is that they are made up of chemical groups whose ionization states are modulated by pH changes that are of the order of 0.4 units. The determination of the protonation states of such chemical groups as a function of conformation of the biomolecule and the pH of the environment can be useful in the elucidation of important biological processes from enzymatic catalysis to protein folding and molecular recognition. In the past 15 years, the theory of Poisson-Boltzmann has been successfully used to estimate the pKa of ionizable sites in proteins yielding results, which may differ by 0.1 unit from the experimental values. In this study, we review the theory of Poisson-Boltzmann under the perspective of its application to the calculation of pKa in proteins.
International Nuclear Information System (INIS)
We apply and compare the preconditioned generalized conjugate gradient methods to solve the linear system equation that arises in the two-dimensional neutron and photon transport equation in this paper. Several subroutines are developed on the basis of preconditioned generalized conjugate gradient methods for time-independent, two-dimensional neutron and photon transport equation in the transport theory. These generalized conjugate gradient methods are used: TFQMR (transpose free quasi-minimal residual algorithm) CGS (conjugate gradient square algorithm), Bi-CGSTAB (bi-conjugate gradient stabilized algorithm) and QMRCGSTAB (quasi-minimal residual variant of bi-conjugate gradient stabilized algorithm). These subroutines are connected to computer program DORT. Several problems are tested on a personal computer with Intel Pentium CPU. The reasons to choose the generalized conjugate gradient methods are that the methods have better residual (equivalent to error) control procedures in the computation and have better convergent rate. The pointwise incomplete LU factorization ILU, modified pointwise incomplete LU factorization MILU, block incomplete factorization BILU and modified blockwise incomplete LU factorization MBILU are the preconditioning techniques used in the several testing problems. In Bi-CGSTAB, CGS, TFQMR and QMRCGSTAB method, we find that either CGS or Bi-CGSTAB method combined with preconditioner MBILU is the most efficient algorithm in these methods in the several testing problems. The numerical solution of flux by preconditioned CGS and Bi-CGSTAB methods has the same result as those from Cray computer, obtained by either the point successive relaxation method or the line successive relaxation method combined with Gaussian elimination
Lattice gas cellular automata and lattice Boltzmann models an introduction
Wolf-Gladrow, Dieter A
2000-01-01
Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Working knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.
Gluon Transport Equation with Effective Mass and Dynamical Onset of Bose-Einstein Condensation
Blaizot, Jean-Paul; Liao, Jinfeng
2015-01-01
We study the transport equation describing a dense system of gluons, in the small scattering angle approximation, taking into account medium-generated effective masses of the gluons. We focus on the case of overpopulated systems that are driven to Bose-Einstein condensation on their way to thermalization. The presence of a mass modifies the dispersion relation of the gluon, as compared to the massless case, but it is shown that this does not change qualitatively the scaling behavior in the vicinity of the onset.
Phase retrieval by using the transport-of-intensity equation with Hilbert transform.
Li, Wei-Shuo; Chen, Chun-Wei; Lin, Kuo-Feng; Chen, Hou-Ren; Tsai, Chih-Ya; Chen, Chyong-Hua; Hsieh, Wen-Feng
2016-04-01
Phase recovery by solving the transport-of-intensity equation (TIE) is a non-iterative and non-interferometric phase retrieval technique. From solving the TIE with conventional, one partial derivative and Hilbert transform methods for both the periodic and aperiodic samples, we demonstrate that the Hilbert transform method can provide the smoother phase images with edge enhancement and fine structures. Furthermore, compared with the images measured by optical and atomic force microscopy, the Hilbert transform method has the ability to quantitatively map out the phase images for both the periodic and aperiodic structures. PMID:27192301
Really TVD advection schemes for the depth-integrated transport equation
Mercier, Ch.; Delhez, E. J. M.
This paper explores the use of TVD advection schemes to solve the depth-integrated transport equation for tracers in finite volume marine models. Numerical experiments show that the blind application of the usual TVD schemes and associated flux limiters can lead to non-TVD solutions when applied in complex geometries. Spatial and/or temporal variations of the local bathymetry can indeed break the TVD property of the usual schemes. Really TVD schemes can be recovered by taking into account the local depth and its variations in the formulation of the flux limiters. Using this approach, a generalized superbee limiter is introduced and validated.
Transport equation theory of electron backscattering and x-ray production
International Nuclear Information System (INIS)
A transport equation theory of electron backscattering and x ray production is derived and applied to energy dissipation of 30-KeV electrons for copper as a function of depth and to the energy distribution of backscattered electrons for copper, aluminum, and gold. These results are plotted and compared with experiment. Plots for variations of backscattering with atomic number and with angle of incidence, and polar plots of backscattering for 30-keV electrons at normal incidence are also presented. 10 references, seven figures
Gluon transport equation with effective mass and dynamical onset of Bose-Einstein condensation
Blaizot, Jean-Paul; Jiang, Yin; Liao, Jinfeng
2016-05-01
We study the transport equation describing a dense system of gluons, in the small scattering angle approximation, taking into account medium-generated effective masses of the gluons. We focus on the case of overpopulated systems that are driven to Bose-Einstein condensation on their way to thermalization. The presence of a mass modifies the dispersion relation of the gluon, as compared to the massless case, but it is shown that this does not change qualitatively the scaling behavior in the vicinity of the onset.
Use of angular finite elements for the numerical analysis of transport equation
International Nuclear Information System (INIS)
In this work, we numerically solve the first order transport equation, by using a finite element method to treat angular variables. In the first part, we show the finite element method continue the usualy used discretization methods. In the second part, we give several properties that the finite element method must satisfy, to solve correctly the problem, in the simple case of monodimensional geometry. In the last part, we compare results obtained with those given by two known codes, in the case of bidimensional geometry
Numerical modeling of photon migration in human neck based on the radiative transport equation
Fujii, Hiroyuki; Nadamoto, Ken; Okada, Eiji; Yamada, Yukio; Hoshi, Yoko; Watanabe, Masao
2016-01-01
Biomedical optical imaging has a possibility of a comprehensive diagnosis of thyroid cancer in conjunction with ultrasound imaging. For improvement of the optical imaging, this study develops a higher order scheme for solving the time-dependent radiative transport equation (RTE) by use of the finite-difference and discrete-ordinate methods. The accuracy and efficiency of the developed scheme are examined by comparison with the analytical solutions of the RTE in homogeneous media. Then, the developed scheme is applied to describing photon migration in the human neck model. The numerical simulations show complex behaviors of photon migration in the human neck model due to multiple diffusive reflection near the trachea.
Application of lattice Boltzmann scheme to nanofluids
Institute of Scientific and Technical Information of China (English)
XUAN Yimin; LI Qiang; YAO Zhengping
2004-01-01
A nanofluid is a particle suspension that consists of base liquids and nanoparticles. Nanofluid has greater potential for heat transfer enhancement than traditional solid-liquid mixture. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles,a lattice Boltzmann model for simulating flow and energy transport processes inside the nanofluids is proposed. The irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids are discussed. The distributions of suspended nanoparticles inside nanofluids are calculated.
Finite-element discretization of 3D energy-transport equations for semiconductors
Energy Technology Data Exchange (ETDEWEB)
Gadau, Stephan
2007-07-01
In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and
Directory of Open Access Journals (Sweden)
M.A.K. Azad
2014-06-01
Full Text Available In this study, the statistical theory of certain distribution functions for simultaneous velocity, magnetic temperature and concentration fields in MHD turbulent flow have been studied. The various properties of the constructed joint distribution functions such as, reduction property, separation property, coincidence and symmetric properties have been discussed. We have made an attempt to derive the transport equations for two and three point distribution functions. Lastly, the transport equation for evaluation of three point distribution functions has been derived.
Energy Technology Data Exchange (ETDEWEB)
Bailey, T S; Chang, J H; Warsa, J S; Adams, M L
2010-12-22
We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretizations that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems.
Energy Technology Data Exchange (ETDEWEB)
Pinchedez, K
1999-06-01
Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)
Topological interactions in a Boltzmann-type framework
Blanchet, Adrien; Degond, Pierre
2015-01-01
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However , it exhibits...
Sengers, J. V.; Basu, R. S.; Sengers, J. M. H. L.
1981-01-01
A survey is presented of representative equations for various thermophysical properties of fluids in the critical region. Representative equations for the transport properties are included. Semi-empirical modifications of the theoretically predicted asymtotic critical behavior that yield simple and practical representations of the fluid properties in the critical region are emphasized.
International Nuclear Information System (INIS)
Transport equation in one-dimensional geometry with linear-anisotropic scattering is written in integral form. Closed system of integral equations relative to current harmonics - φ0(r) and φ1(r) is obtained. The system of integral equations is used as the basis for the PLINTC program. Numerical results on scalar current distribution in cylindrical reactor cells in single-velocity approximation are obtained. Comparison with numerical results obtained through other programs is presented. 10 refs., 1 fig., 3 tabs
Kotake, Kei; Yamada, Shoichi; Takiwaki, Tomoya; Kuroda, Takami; Suwa, Yudai; Nagakura, Hiroki
2012-01-01
This is a status report on our endeavor to reveal the mechanism of core-collapse supernovae (CCSNe) by large-scale numerical simulations. Multi-dimensionality of the supernova engine, general relativisitic magnetohydrodynamics, energy and lepton number transport by neutrinos emitted from the forming neutron star as well as nuclear interactions there, are all believed to play crucial roles in repelling infalling matter and producing energetic explosions. These ingredients are nonlinearly coupled with one another in the dynamics of core-collapse, bounce, and shock expansion. Serious quantitative studies of CCSNe hence make extensive numerical computations mandatory. Since neutrinos are neither in thermal nor in chemical equilibrium in general, their distributions in the phase space should be computed. This is a six dimensional (6D) neutrino transport problem and quite a challenge even for those with an access to the most advanced numerical resources such as the "K computer". To tackle this problem, we have emba...
A numerical spectral approach to solve the dislocation density transport equation
Djaka, K. S.; Taupin, V.; Berbenni, S.; Fressengeas, C.
2015-09-01
A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme.
Lattice Boltzmann modelling of intrinsic permeability
Li, Jun; Wu, Lei; Zhang, Yonghao
2016-01-01
Lattice Boltzmann method (LBM) has been applied to predict flow properties of porous media including intrinsic permeability, where it is implicitly assumed that the LBM is equivalent to the incompressible (or near incompressible) Navier-Stokes equation. However, in LBM simulations, high-order moments, which are completely neglected in the Navier-Stokes equation, are still available through particle distribution functions. To ensure that the LBM simulation is correctly working at the Navier-Stokes hydrodynamic level, the high-order moments have to be negligible. This requires that the Knudsen number (Kn) is small so that rarefaction effect can be ignored. In this technical note, we elaborate this issue in LBM modelling of porous media flows, which is particularly important for gas flows in ultra-tight media.
Analytical solutions of the advection-dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originally established for diffusion type problems, to the case of advective-dispersive transport subj...
Gaeuman, D.; Andrews, E.D.; Kraus, A.; Smith, W.
2009-01-01
Bed load samples from four locations in the Trinity River of northern California are analyzed to evaluate the performance of the Wilcock-Crowe bed load transport equations for predicting fractional bed load transport rates. Bed surface particles become smaller and the fraction of sand on the bed increases with distance downstream from Lewiston Dam. The dimensionless reference shear stress for the mean bed particle size (t*rm) is largest near the dam, but varies relatively little between the more downstream locations. The relation between t*rm and the reference shear stresses for other size fractions is constant across all locations. Total bed load transport rates predicted with the Wilcock-Crowe equations are within a factor of 2 of sampled transport rates for 68% of all samples. The Wilcock-Crowe equations nonetheless consistently under-predict the transport of particles larger than 128 mm, frequently by more than an order of magnitude. Accurate prediction of the transport rates of the largest particles is important for models in which the evolution of the surface grain size distribution determines subsequent bed load transport rates. Values of term estimated from bed load samples are up to 50% larger than those predicted with the Wilcock-Crowe equations, and sampled bed load transport approximates equal mobility across a wider range of grain sizes than is implied by the equations. Modifications to theWilcock-Crowe equation for determining t*rm and the hiding function used to scale term to other grain size fractions are proposed to achieve the best fit to observed bed load transport in the Trinity River. Copyright 2009 by the American eophysical Union.
International Nuclear Information System (INIS)
Highlights: → We developed a variable order global basis scheme to solve light transport in 3D. → Based on finite elements, the method can be applied to a wide class of geometries. → It is computationally cheap when compared to the fixed order scheme. → Comparisons with local basis method and other models demonstrate its accuracy. → Addresses problems encountered n modeling of light transport in human brain. - Abstract: We propose the PN approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the PN approximation is compared against Monte Carlo simulations and other state-of-the-art methods.
Fundamental aspects of plasma chemical physics transport
Capitelli, Mario; Laricchiuta, Annarita
2013-01-01
Fundamental Aspects of Plasma Chemical Physics: Tranpsort develops basic and advanced concepts of plasma transport to the modern treatment of the Chapman-Enskog method for the solution of the Boltzmann transport equation. The book invites the reader to consider actual problems of the transport of thermal plasmas with particular attention to the derivation of diffusion- and viscosity-type transport cross sections, stressing the role of resonant charge-exchange processes in affecting the diffusion-type collision calculation of viscosity-type collision integrals. A wide range of topics is then discussed including (1) the effect of non-equilibrium vibrational distributions on the transport of vibrational energy, (2) the role of electronically excited states in the transport properties of thermal plasmas, (3) the dependence of transport properties on the multitude of Saha equations for multi-temperature plasmas, and (4) the effect of the magnetic field on transport properties. Throughout the book, worked examples ...
Directory of Open Access Journals (Sweden)
Yoshinobu Tanaka
2012-01-01
Full Text Available The overall membrane pair characteristics included in the overall mass transport equation are understandable using the phenomenological equations expressed in the irreversible thermodynamics. In this investigation, the overall membrane pair characteristics (overall transport number , overall solute permeability , overall electro-osmotic permeability and overall hydraulic permeability were measured by seawater electrodialysis changing current density, temperature and salt concentration, and it was found that occasionally takes minus value. For understanding the above phenomenon, new concept of the overall concentration reflection coefficient ∗ is introduced from the phenomenological equation. This is the aim of this investigation. ∗ is defined for describing the permselectivity between solutes and water molecules in the electrodialysis system just after an electric current interruption. ∗ is expressed by the function of and . ∗ is generally larger than 1 and is positive, but occasionally ∗ becomes less than 1 and becomes negative. Negative means that ions are transferred with water molecules (solvent from desalting cells toward concentrating cells just after an electric current interruption, indicating up-hill transport or coupled transport between water molecules and solutes.
Transport coefficients for driven granular mixtures at low-density
Khalil, Nagi; Garzó, Vicente
2013-01-01
The transport coefficients of a granular binary mixture driven by a stochastic bath with friction are determined from the inelastic Boltzmann kinetic equation. A normal solution is obtained via the Chapman-Enskog method for states near homogeneous steady states. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coup...
Hypocoercivity for linear kinetic equations conserving mass
Dolbeault, Jean
2015-02-03
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Directory of Open Access Journals (Sweden)
A. Jaruga
2014-11-01
Full Text Available This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case; and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Directory of Open Access Journals (Sweden)
A. Jaruga
2015-04-01
Full Text Available This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case; and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.
2015-04-01
This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA) on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.
2014-11-01
This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Navier-Stokes transport coefficients of $d$-dimensional granular binary mixtures at low density
Garzo, Vicente; Montanero, Jose Maria
2006-01-01
The Navier-Stokes transport coefficients for binary mixtures of smooth inelastic hard disks or spheres under gravity are determined from the Boltzmann kinetic theory by application of the Chapman-Enskog method for states near the local homogeneous cooling state. It is shown that the Navier-Stokes transport coefficients are not affected by the presence of gravity. As in the elastic case, the transport coefficients of the mixture verify a set of coupled linear integral equations that are approx...
Energy Technology Data Exchange (ETDEWEB)
Lehtikangas, O., E-mail: Ossi.Lehtikangas@uef.fi [Department of Applied Physics, University of Eastern Finland, PO Box 1627, 70211 Kuopio (Finland); Tarvainen, T. [Department of Applied Physics, University of Eastern Finland, PO Box 1627, 70211 Kuopio (Finland); Department of Computer Science, University College London, Gower Street, London WC1E 6BT (United Kingdom); Kim, A.D. [Applied Mathematics Unit, School of Natural Sciences, University of California, Merced, CA 95343 (United States); Arridge, S.R. [Department of Computer Science, University College London, Gower Street, London WC1E 6BT (United Kingdom)
2015-02-01
The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena on the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light.
International Nuclear Information System (INIS)
The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena on the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light
A Transport Equation Approach to Green Functions and Self-force Calculations
Wardell, Barry
2010-01-01
In a recent work, we presented the first application of the Poisson-Wiseman-Anderson method of `matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function which are respectively valid in the `quasilocal' and `distant past' regimes, and which may be matched together within the normal neighbourhood. In this article, we introduce the method of matched expansions and discuss transport equation methods for the calculation of the Green function in the quasilocal region. These methods allow the Green function to be evaluated throughout the normal neighborhood and are also relevant to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity.
Transport equations and linear response of superfluid Fermi mixtures in neutron stars
Gusakov, M E
2010-01-01
We study transport properties of a strongly interacting superfluid mixture of two Fermi-liquids. A typical example of such matter is the neutron-proton liquid in the cores of neutron stars. To describe the mixture, we employ the Landau theory of Fermi-liquids, generalized to allow for the effects of superfluidity. We formulate the kinetic equation and analyze linear response of the system to vector (e.g., electromagnetic) perturbation. In particular, we calculate the transverse and longitudinal polarization functions for both liquid components. We demonstrate, that they can be expressed through the Landau parameters of the mixture and polarization functions of noninteracting matter (when the Landau quasiparticle interaction is neglected). Our results can be used, e.g., for studies of the kinetic coefficients and low-frequency long-wavelength collective modes in superfluid Fermi-mixtures.
Zhang, Chenggong; He, Wenqi; Wu, Jiachen; Peng, Xiang
2015-04-01
A novel optical cryptosystem based on phase-truncated Fresnel diffraction (PTFD) and transport of intensity equation (TIE) is proposed. By using the phase truncation technique, a phase-encoded plaintext could be encrypted into a real-valued noise-like intensity distribution by employing a random amplitude mask (RAM) and a random phase mask (RPM), which are regarded as two secret keys. For decryption, a generalized amplitude-phase retrieval (GAPR) algorithm combined with the TIE method are proposed to recover the plaintext with the help of two keys. Different from the current phase-truncated-based optical cryptosystems which need record the truncated phase as decryption keys, our scheme do not need the truncated phase because of the introducing of the TIE method. Moreover, the proposed scheme is expected to against existing attacks. A set of numerical simulation results show the feasibility and security of the proposed method. PMID:25968722
Silicon wafer microstructure imaging using InfraRed Transport of Intensity Equation
Li, Hongru; Feng, Guoying; Bourgade, Thomas; Zuo, Chao; Du, Yongzhao; Zhou, Shouhuan; Asundi, Anand
2015-03-01
A novel quantitative 3D imaging system of silicon microstructures using InfraRed Transport of Intensity Equation (IRTIE) is proposed in this paper. By recording the intensity at multiple planes and using FFT or DCT based TIE solver, fast and accurate phase retrieval for both uniform and non-uniform intensity distributions is proposed. Numerical simulation and experiments confirm the accuracy and reliability of the proposed method. The application of IR-TIE for inspection of micro-patterns in visibly opaque media using 1310 nm light source is demonstrated. For comparison, micro-patterns are also inspected by the contact scanning mode Taylor Hobson system. Quantitative agreement suggests the possibility of using IR-TIE for phase imaging of silicon wafers.
Directory of Open Access Journals (Sweden)
Xia Wang
2010-01-01
Full Text Available An interfacial area transport equation (IATE, proposed to dynamically describe the interfacial structure evolution of two-phase flows, could help improve the predictive capability of the two-fluid model. The present study aims to investigate the well-posedness issue of a one-dimensional two-fluid model with the IATE (named “two-fluid-IATE model” hereafter using a characteristic analysis. The momentum flux parameters, which take into account the coupling of the volumetric fraction of phase and velocity distributions over the cross-section of a flow passage, are employed. A necessary condition for the system to achieve hyperbolicity under an adiabatic flow condition is identified. A case study is performed for an adiabatic liquid-liquid slug flow, which shows that the hyperbolicity of the two-fluid-IATE model is guaranteed if appropriate correlations of the momentum flux parameters are applied in the two-fluid-IATE model.
DEFF Research Database (Denmark)
Svec, Oldrich; Skoček, Jan
2013-01-01
The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary conditi...
Master equation approach to charge injection and transport in organic insulators.
Freire, José A; Voss, Grasiela
2005-03-22
We develop a master equation model of a disordered organic insulator sandwiched between metallic electrodes by treating as rate processes both the injection and the internal transport. We show how the master equation model allows for the inclusion of crucial correlation effects in the charge transport, particularly of the Pauli exclusion principle and of space-charge effects, besides, being dependent on just the microscopic form of the transfer rate between the localized electronic states, it allows for the investigation of different microscopic scenarios in the organic, such as polaronic hopping, correlated energy levels, interaction with image charge, etc. The model allows for a separate analysis of the injection and the recombination currents. We find that the disorder, besides increasing the injection current, eliminates the possibility of observation of a Fowler-Nordheim injection current at zero temperature, and that it does not alter the Schottky barrier size of the zero-field thermionic injection current from the value based on the energy difference between the electrode Fermi level and the highest occupied molecular orbital/lowest unoccupied molecular orbital levels in the organic, but it makes the Arrhenius temperature dependence appear at larger temperatures. We investigate how the I(V) characteristics of a device is affected by the presence of correlations in the site energy distribution and by the form of the internal hopping rate, specifically the Miller-Abrahams rate and the Marcus or small-polaron rate. We show that the disorder does not modify significantly the ebeta square root E field dependence of the net current due to the Schottky barrier lowering caused by the attraction between the charge and its image in the electrode. PMID:15836407
Interfacial area transport equation of gas-liquid two-phase flow across a horizontal tube
International Nuclear Information System (INIS)
It is important to improve accuracy of the two-phase flow simulation codes, since these codes are used for the performance estimation and the safety designs of the industrial products which two-phase flow occur inside. In the governing equations of the two-phase flow simulation model, interfacial area concentration is one of the most important parameters to improve the model. Some researcher developed a transport equation of the interfacial area concentration that include source and sink terms due to bubble coalescence and breakup. However, these equations use numerous parameters to obtain the source and sink terms. In this study, we have successfully expressed the source and sink term as functions of a single variable to simplify and enhance their usefulness. The coalescence of bubbles was expressed as a function of the local void fraction, and the breakup of bubbles was expressed as a function of the time averaged local liquid velocity. Using this model, we can simulate the local void fraction and interfacial area concentration. In order to validate the simulation model, we also carried out experiments to measure a time averaged local interfacial area concentration and a time averaged local void fraction in a vertical upward air-water two-phase flow across horizontal tube with a double sensor electrical conductivity probe. The test section comprised a 50mm × 50 mm square channel partially obstructed by a single horizontal tube inside. The superficial gas velocity and superficial liquid velocity ranged from 0.019 to 0.215 m/s and from 0.133 to 0.533 m/s, respectively. Measurements of the void fraction and the interfacial area concentration were performed at three axial locations of z = 110, 340, and 640 mm and transverse locations from r = 0 to 25 mm. The experimental data set were compared to the result of the numerical simulations. Good agreement is obtained between these results. (author)
Joint Training of Deep Boltzmann Machines
Goodfellow, Ian; Courville, Aaron; Bengio, Yoshua
2012-01-01
We introduce a new method for training deep Boltzmann machines jointly. Prior methods require an initial learning pass that trains the deep Boltzmann machine greedily, one layer at a time, or do not perform well on classifi- cation tasks.
Development of a Prototype Lattice Boltzmann Code for CFD of Fusion Systems.
Energy Technology Data Exchange (ETDEWEB)
Pattison, Martin J; Premnath, Kannan N; Banerjee, Sanjoy; Dwivedi, Vinay
2007-02-26
ability of the LBM to scale almost linearly. The equation for magnetic induction was also solved using a lattice Boltzmann method. This approach has the advantage that it fits in well to the framework used for the hydrodynamic equations, but more importantly that it preserves the ability of the code to run efficiently on parallel architectures. Since the LBM is a relatively recent model, a number of new developments were needed to solve the magnetic induction equation for practical problems. Existing methods were only suitable for cases where the fluid viscosity and the magnetic resistivity are of the same order, and a preconditioning method was used to allow the simulation of liquid metals, where these properties differ by several orders of magnitude. An extension of this method to the hydrodynamic equations allowed faster convergence to steady state. A new method of imposing boundary conditions using an extrapolation technique was derived, enabling the magnetic field at a boundary to be specified. Also, a technique by which the grid can be stretched was formulated to resolve thin layers at high imposed magnetic fields, allowing flows with Hartmann numbers of several thousand to be quickly and efficiently simulated. In addition, a module has been developed to calculate the temperature field and heat transfer. This uses a total variation diminishing scheme to solve the equations and is again very amenable to parallelisation. Although, the module was developed with thermal modelling in mind, it can also be applied to passive scalar transport. The code is fully three dimensional and has been applied to a wide variety of cases, including both laminar and turbulent flows. Validations against a series of canonical problems involving both MHD effects and turbulence have clearly demonstrated the ability of the LBM to properly model these types of flow. As well as applications to fusion engineering, the resulting code is flexible enough to be applied to a wide range of other
Egami, Yoshiyuki; Iwase, Shigeru; Tsukamoto, Shigeru; Ono, Tomoya; Hirose, Kikuji
2015-09-01
We develop a first-principles electron-transport simulator based on the Lippmann-Schwinger (LS) equation within the framework of the real-space finite-difference scheme. In our fully real-space-based LS (grid LS) method, the ratio expression technique for the scattering wave functions and the Green's function elements of the reference system is employed to avoid numerical collapse. Furthermore, we present analytical expressions and/or prominent calculation procedures for the retarded Green's function, which are utilized in the grid LS approach. In order to demonstrate the performance of the grid LS method, we simulate the electron-transport properties of the semiconductor-oxide interfaces sandwiched between semi-infinite jellium electrodes. The results confirm that the leakage current through the (001)Si-SiO_{2} model becomes much larger when the dangling-bond state is induced by a defect in the oxygen layer, while that through the (001)Ge-GeO_{2} model is insensitive to the dangling bond state. PMID:26465580
Tian, Xiaolin; Meng, Xin; Yu, Wei; Song, Xiaojun; Xue, Liang; Liu, Cheng; Wang, Shouyu
2016-10-01
Microscopy combined with the transport of intensity equation is capable of retrieving both intensity and phase distributions of samples from both in-focus and defocus intensities. However, during measurements, the focal plane is often decided artificially and the improper choice may induce errors in quantitative intensity and phase retrieval. In order to obtain accurate in-focus information, quantitative intensity and phase imaging with the numerical focusing transport of intensity equation method combined with cellular duty ratio criterion and numerical wavefront propagation is introduced in this paper. Both numerical simulations and experimental measurements are provided proving this designed method can increase both retrieved in-focus intensity and phase accuracy and reduce dependence of focal plane determination in transport of intensity equation measurements. It is believed that the proposed method can be potentially applied in various fields as in-focus compensation for quantitative phase imaging and automatic focal plane determination, etc.
Energy Technology Data Exchange (ETDEWEB)
Azmy, Yousry
2014-06-10
We employ the Integral Transport Matrix Method (ITMM) as the kernel of new parallel solution methods for the discrete ordinates approximation of the within-group neutron transport equation. The ITMM abandons the repetitive mesh sweeps of the traditional source iterations (SI) scheme in favor of constructing stored operators that account for the direct coupling factors among all the cells' fluxes and between the cells' and boundary surfaces' fluxes. The main goals of this work are to develop the algorithms that construct these operators and employ them in the solution process, determine the most suitable way to parallelize the entire procedure, and evaluate the behavior and parallel performance of the developed methods with increasing number of processes, P. The fastest observed parallel solution method, Parallel Gauss-Seidel (PGS), was used in a weak scaling comparison with the PARTISN transport code, which uses the source iteration (SI) scheme parallelized with the Koch-baker-Alcouffe (KBA) method. Compared to the state-of-the-art SI-KBA with diffusion synthetic acceleration (DSA), this new method- even without acceleration/preconditioning-is completitive for optically thick problems as P is increased to the tens of thousands range. For the most optically thick cells tested, PGS reduced execution time by an approximate factor of three for problems with more than 130 million computational cells on P = 32,768. Moreover, the SI-DSA execution times's trend rises generally more steeply with increasing P than the PGS trend. Furthermore, the PGS method outperforms SI for the periodic heterogeneous layers (PHL) configuration problems. The PGS method outperforms SI and SI-DSA on as few as P = 16 for PHL problems and reduces execution time by a factor of ten or more for all problems considered with more than 2 million computational cells on P = 4.096.
Fiorentino, Eve-Agnès; Toussaint, Renaud; Jouniaux, Laurence
2014-05-01
We study the coupling between hydraulic and electric flows in a porous medium at small scale using the Lattice Boltzmann method. This method is a computational fluid dynamics technique that is used for advection and diffusion modeling. We implement a coupled Lattice Boltzmann algorithm that solves both the mass transport and the electric field arising from charges displacements. The streaming potential and electroosmosis phenomena occur in a variety of situations and derive from this coupling. We focus on the streaming potential which is described using the ratio between the created potential difference and the applied pressure gradient. The streaming potential is assumed to be a linear function of the fluid conductivity, but experimental results highlight anomalous behaviors at low and high salinity. We try to account for them by setting extreme conditions that are likely to generate non-linearities. Several pore radii are tested so as to determine what is the effect of a radius that is comparable to the Debye length, the screening length of the electric potential, due to the ions in the electrolyte. The volumetric integral of the electrical current is calculated for comparison with the 2D simulations. High values of zeta potential are tested to verify if the discrepancy regarding the theoretical result is concentration-dependent. We try to include a surface conductivity term in the coefficient formulation. Some tests including a rugosity on the channel walls are performed. All of these attempts show a normal behaviour of the streaming potential at high salinity. We observe a decrease of the ratio at low conductivity, showing that this ratio is modified when the pore radius becomes negligible compared with the Debye length, which is physically meaningful in little pores at low concentrations. References : S. Pride. Governing equations for the coupled electromagnetics and acoustics of porous media. Physical Review B, 50 : 15678-15696, 1994. D. A. Wolf
Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio
Indian Academy of Sciences (India)
Tao Jiang; Qiwei Gong; Ruofan Qiu; Anlin Wang
2014-10-01
An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, an interaction potential between particles is incorporated into the discrete lattice Boltzmann equation through the equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce `spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.
Romano, Vittorio; Majorana, Armando; Coco, Marco
2015-12-01
A new algorithm for Monte Carlo simulations of charge transport in semiconductors is devised in order to properly deal with Pauli's exclusion principle in the degenerate case. Applications are presented in the case of monolayer graphene and comparisons with solutions of the Boltzmann equation obtained by using a discontinuous Galerkin method furnish a cross-validation of the proposed approach.
Radiation Transport Calculations and Simulations
Energy Technology Data Exchange (ETDEWEB)
Fasso, Alberto; /SLAC; Ferrari, A.; /CERN
2011-06-30
This article is an introduction to the Monte Carlo method as used in particle transport. After a description at an elementary level of the mathematical basis of the method, the Boltzmann equation and its physical meaning are presented, followed by Monte Carlo integration and random sampling, and by a general description of the main aspects and components of a typical Monte Carlo particle transport code. In particular, the most common biasing techniques are described, as well as the concepts of estimator and detector. After a discussion of the different types of errors, the issue of Quality Assurance is briefly considered.
Multi-component lattice-Boltzmann model with interparticle interaction
Shan, Xiaowen; Doolen, Gary
1995-01-01
A previously proposed [X. Shan and H. Chen, Phys. Rev. E {\\bf 47}, 1815, (1993)] lattice Boltzmann model for simulating fluids with multiple components and interparticle forces is described in detail. Macroscopic equations governing the motion of each component are derived by using Chapman-Enskog method. The mutual diffusivity in a binary mixture is calculated analytically and confirmed by numerical simulation. The diffusivity is generally a function of the concentrations of the two component...
A Lattice Boltzmann model for diffusion of binary gas mixtures
Bennett, Sam
2010-01-01
This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth. The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the sca...
Lattice Boltzmann Method for mixtures at variable Schmidt number
Monteferrante, Michele; Melchionna, Simone; Marconi, Umberto Marini Bettolo
2015-01-01
When simulating multicomponent mixtures via the Lattice Boltzmann Method, it is desirable to control the mutual diffusivity between species while maintaining the viscosity of the solution fixed. This goal is herein achieved by a modification of the multicomponent Bhatnagar-Gross-Krook (BGK) evolution equations by introducing two different timescales for mass and momentum diffusion. Diffusivity is thus controlled by an effective drag force acting between species. Numerical simulations confirm ...
Stochastic particle approximations for generalized Boltzmann models and convergence estimates
Graham, Carl; Méléard, Sylvie
1997-01-01
We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniqu...
Energy Technology Data Exchange (ETDEWEB)
Carella, Alfredo Raul
2012-09-15
Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)
Ludwig Boltzmann A Pioneer of Modern Physics
Flamm, D
1997-01-01
In two respects Ludwig Boltzmann was a pioneer of quantum mechanics. First because in his statistical interpretation of the second law of thermodynamics he introduced the theory of probability into a fundamental law of physics and thus broke with the classical prejudice, that fundamental laws have to be strictly deterministic. Even Max Planck had not been ready to accept Boltzmann's statistical methods until 1900. With Boltzmann's pioneering work the probabilistic interpretation of quantum mechanics had already a precedent. In fact in a paper in 1897 Boltzmann had already suggested to Planck to use his statistical methods for the treatment of black body radiation. The second pioneering step towards quantum mechanics was Boltzmann's introduction of discrete energy levels. Boltzmann used this method already in his 1872 paper on the H-theorem. One may ask whether Boltzmann considered this procedure only as a mathematical device or whether he attributed physical significance to it. In this connection Ostwald repo...
Analysis and development of spatial hp-refinement methods for solving the neutron transport equation
International Nuclear Information System (INIS)
The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4. generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called SN approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of hp-refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into sub-cells, or by order refinement (p-refinement), by increasing the order of the polynomial basis. In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores. These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
Degond, Pierre; Savelief, Dominique; Vignal, Marie-Hélène
2010-01-01
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately.
Barraud, Sylvain
2009-09-01
Various theoretical formulations are proposed for investigating the carrier transport in nanoscale electronic devices. In this paper, a discrete formulation of the Wigner transport equation (WTE) for the self-consistent simulation of phase-coherent quantum transport in silicon nanowire metal-oxide-semiconductor field-effect transistor (MOSFET) devices is presented. The device is simulated using an effective-mass Hamiltonian within the mode-space approximation. The numerical scheme proposed in this work solves self-consistently three dimensional Poisson's equation, two dimensional Schrödinger's equation in each cross-sectional plane of the nanowire, and the steady-state one dimensional WTE for each conduction mode to handle the quantum transport along the channel. Details on numerical implementation of the Wigner function method are given, and the results are compared with those of the nonequilibrium Green's function (NEGF) method in the ballistic limit. The calculations of current-voltage electrical characteristics of surround-gated silicon nanowires are performed using both NEGF and WTE formulations. The good agreement observed between these approaches means that a direct solution of the WTE is an accurate simulation method for modeling the ballistic quantum transport in silicon nanowire MOSFETs.
Indian Academy of Sciences (India)
Marko Žnidarič
2011-11-01
We discuss recent ﬁndings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to ﬁnd a stationary solution of Lindblad master equation. Furthermore, for a speciﬁc model an exact solution is presented.
Institute of Scientific and Technical Information of China (English)
Liming WU; Zhengliang ZHANG
2006-01-01
We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to ReactionDiffusion equations are provided.
International Nuclear Information System (INIS)
One of the main problems in reactor physics is to determine the neutron distribution in reactor core, since knowing that, it is possible to calculate the rapidity of occurrence of different nuclear reaction inside the reactor core. Within different theories existing in nuclear reactor physics, is neutron transport the one in which equation who govern the exact behavior of neutronic distribution are developed even inside the proper neutron transport theory, there exist different methods of solution which are approximations to exact solution; still more, with the purpose to reach a more precise solution, the majority of methods have been approached to the obtention of solutions in numerical form with the aim of take the advantages of modern computers, and for this reason a great deal of effort is dedicated to numerical solution of the equations of neutron transport. In agreement with the above mentioned, in this work has been developed a computer program which uses a relatively new techniques known as 'acceleration of synthetic diffusion' which has been applied to solve the neutron transport equation with 'classical schemes of spatial integration' obtaining results with a smaller quantity of interactions, if they compare to done without using such equation (Author)
Langel, Christopher Michael
A computational investigation has been performed to better understand the impact of surface roughness on the flow over a contaminated surface. This thesis highlights the implementation and development of the roughness amplification model in the flow solver OVERFLOW-2. The model, originally proposed by Dassler, Kozulovic, and Fiala, introduces an additional scalar field roughness amplification quantity. This value is explicitly set at rough wall boundaries using surface roughness parameters and local flow quantities. This additional transport equation allows non-local effects of surface roughness to be accounted for downstream of rough sections. This roughness amplification variable is coupled with the Langtry-Menter model and used to modify the criteria for transition. Results from flat plate test cases show good agreement with experimental transition behavior on the flow over varying sand grain roughness heights. Additional validation studies were performed on a NACA 0012 airfoil with leading edge roughness. The computationally predicted boundary layer development demonstrates good agreement with experimental results. New tests using varying roughness configurations are being carried out at the Texas A&M Oran W. Nicks Low Speed Wind Tunnel to provide further calibration of the roughness amplification method. An overview and preliminary results are provided of this concurrent experimental investigation.
Energy Technology Data Exchange (ETDEWEB)
Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States); Walker, H.F. [Utah State Univ., Logan, UT (United States). Dept. of Mathematics and Statistics
1997-02-01
The solution of the governing steady transport equations for momentum, heat and mass transfer in flowing fluids can be very difficult. These difficulties arise from the nonlinear, coupled, nonsymmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this manuscript the authors focus on evaluating a proposed nonlinear solution method based on an inexact Newton method with backtracking. In this context they use a particular spatial discretization based on a pressure stabilized Petrov-Galerkin finite element formulation of the low Mach number Navier-Stokes equations with heat and mass transport. The discussion considers computational efficiency, robustness and some implementation issues related to the proposed nonlinear solution scheme. Computational results are presented for several challenging CFD benchmark problems as well as two large scale 3D flow simulations.
A least-squares finite-element S{sub n} method for solving first-order neutron transport equation
Energy Technology Data Exchange (ETDEWEB)
Ju Haitao [School of Energy and Power Engineering, Xi' an Jiaotong University, Xi' an 710049 (China)]. E-mail: jht0@hotmail.com; Wu Hongchun [School of Energy and Power Engineering, Xi' an Jiaotong University, Xi' an 710049 (China); Zhou Yongqiang [School of Energy and Power Engineering, Xi' an Jiaotong University, Xi' an 710049 (China); Cao Liangzhi [School of Energy and Power Engineering, Xi' an Jiaotong University, Xi' an 710049 (China); Yao Dong [Nuclear Power Institute of China, Chengdu 610041 (China); Xian, Chun-Yu [Nuclear Power Institute of China, Chengdu 610041 (China)
2007-04-15
A discrete ordinates finite-element method for solving the two-dimensional first-order neutron transport equation is derived using the least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation which contains the inversion of the cross-section. Different from using the standard Galerkin variation to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. To eliminate the discontinuity of the angular flux on the vacuum boundary in the spherical harmonics method, the angle variable is discretized by the discrete ordinates method. A two-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results verified the validity of this method.
Directory of Open Access Journals (Sweden)
Kulish Vladimir V.
2004-01-01
Full Text Available This paper presents an integral solution of the generalized one-dimensional equation of energy transport with the convective term.The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of noninteger orders. Confluent hypergeometric functions, known as Whittaker's functions, appear in the course of the solution procedure upon applying the Laplace transform to the original transport equation.The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the transported property (e.g., temperature, mass, momentum, etc. and its flux.The solution is valid everywhere within the domain, including the domain boundary.
Multiphase cascaded lattice Boltzmann method
Lycett-Brown, D.; Luo, K. H.
2014-01-01
To improve the stability of the lattice Boltzmann method (LBM) at high Reynolds number the cascaded LBM has recently been introduced. As in the multiple relaxation time (MRT) method the cascaded LBM introduces additional relaxation times into the collision operator, but does so in a co-moving reference frame. This has been shown to significantly increase stability at low viscosity in the single phase case. Here the cascaded LBM is further developed to include multiphase flow. For this the for...
A shallow water equation solver and particle tracking method to evaluate the debris transport
International Nuclear Information System (INIS)
Debris generated by loss-of-coolant accident (LOCA) may run all over the containment floor, block the sump screen (or strainer), increase the hydraulic head loss across the screen, and eventually, have an adverse effect to long term recirculation cooling operation in pressurized water reactor (PWR). To resolve the problem from the issue, the replacement of containment recirculation sump strainer is being performed for the most of operating nuclear power plants (NPP) having limited strainer areas. The screen area required to incorporate the potential debris loading has been determined using the transport fraction (TF) defined by a ratio of amount of debris accumulated on screen to one generated by LOCA. For the most conventional NPP, the debris transport to the sump screen is initiated by the recirculation actuation. Therefore, evaluation of TF was based on the separated analyses on how the debris generated by LOCA is distributed to the containment floor before recirculation and on how much debris is transported by the flow in the containment pool after recirculation, respectively. This led to an approach to obtain TF values for blowdown phase, wash-down phase, pool recirculation phase, separately. Especially, the TF during recirculation phase has been calculated by steady state CFD analysis based that the break flow and recirculation safety injection flow are balanced, thus steady state flow field over the containment is established. However, such a phase separation cannot be applied to some NPP like the Advanced Power Reactor 1400 (APR1400) having no recirculation operation. Transport of debris to sump in the APR1400 is initiated from the early phase of a LOCA in fully transient manner. The present study is to calculate debris transport on the containment floor to sump in APR1400. For this purpose, a hydraulic model to calculate the transient flow field and a particle tracking model to trace the debris particle within the calculation domain are discussed
High order fluid model for streamer discharges: I. Derivation of model and transport data
Dujko, S; White, R D; Ebert, U
2013-01-01
Streamer discharges pose basic problems in plasma physics, as they are very transient, far from equilibrium and have high ionization density gradients; they appear in diverse areas of science and technology. The present paper focuses on the derivation of a high order fluid model for streamers. Using momentum transfer theory, the fluid equations are obtained as velocity moments of the Boltzmann equation; they are closed in the local mean energy approximation and coupled to the Poisson equation for the space charge generated electric field. The high order tensor in the energy flux equation is approximated by the product of two lower order moments to close the system. The average collision frequencies for momentum and energy transfer in elastic and inelastic collisions for electrons in molecular nitrogen are calculated from a multi term Boltzmann equation solution. We then discuss, in particular, (1) the correct implementation of transport data in streamer models; (2) the accuracy of the two term approximation f...
An integrated Boltzmann+hydrodynamics approach to heavy ion collisions
Energy Technology Data Exchange (ETDEWEB)
Petersen, Hannah
2009-04-22
In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. Predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies. Studies of phase diagram trajectories using hydrodynamics are performed. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The full (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. Three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent v{sub 2} values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from E{sub lab}=2-160 A GeV. The HBT correlation of the negatively charged pion source
International Nuclear Information System (INIS)
A coupled (hybrid) simulation strategy spatially combining the finite volume method (FVM) and the lattice Boltzmann method (LBM), called CFVLBM, is developed to simulate coupled multi-scale multi-physicochemical processes. In the CFVLBM, computational domain of multi-scale problems is divided into two sub-domains, i.e., an open, free fluid region and a region filled with porous materials. The FVM and LBM are used for these two regions, respectively, with information exchanged at the interface between the two sub-domains. A general reconstruction operator (RO) is proposed to derive the distribution functions in the LBM from the corresponding macro scalar, the governing equation of which obeys the convection–diffusion equation. The CFVLBM and the RO are validated in several typical physicochemical problems and then are applied to simulate complex multi-scale coupled fluid flow, heat transfer, mass transport, and chemical reaction in a wall-coated micro reactor. The maximum ratio of the grid size between the FVM and LBM regions is explored and discussed. -- Highlights: •A coupled simulation strategy for simulating multi-scale phenomena is developed. •Finite volume method and lattice Boltzmann method are coupled. •A reconstruction operator is derived to transfer information at the sub-domains interface. •Coupled multi-scale multiple physicochemical processes in micro reactor are simulated. •Techniques to save computational resources and improve the efficiency are discussed
Elliptic equation for random walks. Application to transport in microporous media
DEFF Research Database (Denmark)
Shapiro, Alexander
2007-01-01
We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes...... into account the distribution of the residence times of molecules ill pores. The new elliptic diffusion equation is strictly derived by the operator approach. A criterion showing where the new equation should be applied instead of the standard diffusion equation is obtained. Boundary conditions are studied...
Student understanding of the Boltzmann factor
Smith, Trevor I; Thompson, John R
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable, nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations on student discussions about the Boltzmann f...
Master equation for a quantum particle in a gas
Hornberger, Klaus
2006-01-01
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts non-perturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as li...
Quadrature-based Lattice Boltzmann Model for Relativistic Flows
Blaga, Robert
2016-01-01
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.
Application of the three-dimensional telegraph equation to cosmic-ray transport
Tautz, R C
2016-01-01
An analytical solution to the the three-dimensional telegraph equation is presented. This equation has recently received some attention but so far the treatment has been one-dimensional. By using the structural similarity to the Klein-Gordon equation, the telegraph equation can be solved in closed form. Illustrative examples are used to discuss the qualitative differences to the diffusion solution. The comparison with a numerical test-particle simulation reveals that some features of an intensity profile can be better explained using the telegraph approach.
Relativistic quantum transport coefficients for second-order viscous hydrodynamics
Florkowski, Wojciech; Jaiswal, Amaresh; Maksymiuk, Ewa; Ryblewski, Radoslaw; Strickland, Michael
2015-01-01
We express the transport coefficients appearing in the second-order evolution equations for bulk viscous pressure and shear stress tensor using Bose-Einstein, Boltzmann, and Fermi-Dirac statistics for the equilibrium distribution function and Grad's 14-moment approximation as well as the method of Chapman-Enskog expansion for the non-equilibrium part. Specializing to the case of transversally homogeneous and boost-invariant longitudinal expansion of the viscous medium, we compare the results ...
International Nuclear Information System (INIS)
The phase space finite element method when applied to the first order transport equation results in a matrix equation that can only be solved directly. While the direct solution procedure is competitive with more established transport methods for one-dimensional geometries, it is much too slow in two or more dimensions. This summary presents two alternative implementations of the finite element method. Both reduce the size of the matrix to be solved by dividing the spatial domain into smaller regions, and both realize increased efficiency in multi-dimensional applications. Following brief derivations of the methods, computational results for both one-dimensional and two-dimensional applications will be presented and compared for the two distinct approaches
Energy Technology Data Exchange (ETDEWEB)
Clouet, J.F.; Samba, G. [CEA Bruyeres-le-Chatel, 91 (France)
2005-07-01
We use asymptotic analysis to study the diffusion limit of the Symbolic Implicit Monte-Carlo (SIMC) method for the transport equation. For standard SIMC with piecewise constant basis functions, we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation. Nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution. This improvement allows the calculation in opaque medium on a mesh resolving the diffusion scale much larger than the transport scale. Anyway, the huge number of particles which is necessary to get a correct answer makes this computation time consuming. Thus, we have derived from this asymptotic study an hybrid method coupling deterministic calculation in the opaque medium and Monte-Carlo calculation in the transparent medium. This method gives exactly the same results as the previous one but at a much lower price. We present numerical examples which illustrate the analysis. (authors)
Lattice Boltzmann method for bosons and fermions and the fourth order Hermite polynomial expansion
Coelho, Rodrigo C V; Doria, M M; Pereira, R M; Aibe, Valter Yoshihiko
2013-01-01
The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried until the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by J.Y. Yang et al through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded until fourth order in the Hermite polynomials.